# 9.E: Exercises

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## Exercise $$\PageIndex{1}$$

Suppose you have $$\mathbb{R}^2$$ and the $$+$$ operation is as follows: $(a,b) + (c,d) = (a+d,b+c).\nonumber$ Scalar multiplication is defined in the usual way. Is this a vector space? Explain why or why not.

## Exercise $$\PageIndex{2}$$

Suppose you have $$\mathbb{R}^2$$ and the $$+$$ operation is as follows: $(a,b) + (c,d) = (0,b+d)\nonumber$ Scalar multiplication is defined in the usual way. Is this a vector space? Explain why or why not.

## Exercise $$\PageIndex{3}$$

Suppose you have $$\mathbb{R}^2$$ and scalar multiplication is defined as $$c(a,b) = (a, cb)$$ while vector addition is defined as usual. Is this a vector space? Explain why or why not.

## Exercise $$\PageIndex{4}$$

Suppose you have $$\mathbb{R}^2$$ and the $$+$$ operation is defined as follows. $(a,b) + (c,d) = (a−c,b−d)\nonumber$ Scalar multiplication is same as usual. Is this a vector space? Explain why or why not.

## Exercise $$\PageIndex{5}$$

Consider all the functions defined on a non empty set which have values in $$\mathbb{R}$$. Is this a vector space? Explain. The operations are defined as follows. Here $$f ,g$$ signify functions and $$a$$ is a scalar \begin{aligned} (f+g)(x)&=f(x)+g(x) \\ (af)(x)&=a(f(x))\end{aligned}

## Exercise $$\PageIndex{6}$$

Denote by $$\mathbb{R}^{\mathbb{N}}$$ the set of real valued sequences. For $$\vec{a} ≡ \{a_n\}_{n=1}^∞$$, $$\vec{b} ≡ \{b_n\}_{n=1}^\infty$$ two of these, define their sum to be given by $\vec{a}+\vec{b}=\{a_n+b_n\}_{n=1}^\infty\nonumber$ and define scalar multiplication by $c\vec{a}=\{ca_n\}_{n=1}^\infty\text{ where }\vec{a}=\{a+n\}_{n=1}^\infty\nonumber$ Is this a special case of Exercise $$\PageIndex{5}$$? Is this a vector space?

## Exercise $$\PageIndex{7}$$

Let $$\mathbb{C}^2$$ be the set of ordered pairs of complex numbers. Define addition and scalar multiplication in the usual way. $(z,w) + (\hat{z},\hat{w}) = (z+\hat{z},w+\hat{w}), u(z,w) ≡ (uz,uw)\nonumber$ Here the scalars are from $$\mathbb{C}$$. Show this is a vector space.

## Exercise $$\PageIndex{8}$$

Let $$V$$ be the set of functions defined on a nonempty set which have values in a vector space $$W$$. Is this a vector space? Explain.

## Exercise $$\PageIndex{9}$$

Consider the space of $$m\times n$$ matrices with operation of addition and scalar multiplication defined the usual way. That is, if $$A,B$$ are two $$m\times n$$ matrices and $$c$$ a scalar, $(A+B)_{ i j} = A_{i j} +B_{i j}, \:(cA)_{ i j} ≡ c (A_{ij})\nonumber$

## Exercise $$\PageIndex{10}$$

Consider the set of $$n\times n$$ symmetric matrices. That is, $$A = A^T$$. In other words, $$A_{i j} = A_{ji}$$. Show that this set of symmetric matrices is a vector space and a subspace of the vector space of $$n\times n$$ matrices.

## Exercise $$\PageIndex{11}$$

Consider the set of all vectors in $$\mathbb{R}^2 ,(x, y)$$ such that $$x + y ≥ 0$$. Let the vector space operations be the usual ones. Is this a vector space? Is it a subspace of $$\mathbb{R}^2$$?

## Exercise $$\PageIndex{12}$$

Consider the vectors in $$\mathbb{R}^2 ,(x, y)$$ such that $$xy = 0$$. Is this a subspace of $$\mathbb{R}^2$$? Is it a vector space? The addition and scalar multiplication are the usual operations.

## Exercise $$\PageIndex{13}$$

Define the operation of vector addition on $$\mathbb{R}^2$$ by $$(x, y) + (u, v) = (x+u, y+v+1)$$. Let scalar multiplication be the usual operation. Is this a vector space with these operations? Explain.

## Exercise $$\PageIndex{14}$$

Let the vectors be real numbers. Define vector space operations in the usual way. That is $$x+y$$ means to add the two numbers and $$xy$$ means to multiply them. Is $$\mathbb{R}$$ with these operations a vector space? Explain.

## Exercise $$\PageIndex{15}$$

Let the scalars be the rational numbers and let the vectors be real numbers which are the form $$a+b\sqrt{2}$$ for $$a,b$$ rational numbers. Show that with the usual operations, this is a vector space.

## Exercise $$\PageIndex{16}$$

Let $$\mathbb{P}_2$$ be the set of all polynomials of degree $$2$$ or less. That is, these are of the form $$a+bx+cx^2$$. Addition is defined as $(a+bx+cx^2)+(\hat{d}+\hat{b}x+\hat{c}x^2)=(a+\hat{a})+(b+\hat{b})x+(c+\hat{c})x^2\nonumber$ and scalar multiplication is defined as $d(a+bx+cx^2)=da+dbx+cdx^2\nonumber$ Show that, with this definition of the vector space operations that $$\mathbb{P}_2$$ is a vector space. Now let $$V$$ denote those polynomials $$a+bx+cx^2$$ such that $$a+b+c = 0$$. Is $$V$$ a subspace of $$\mathbb{P}_2$$? Explain.

## Exercise $$\PageIndex{17}$$

Let $$M,N$$ be subspaces of a vector space $$V$$ and consider $$M +N$$ defined as the set of all $$m+n$$ where $$m ∈ M$$ and $$n ∈ N$$. Show that $$M +N$$ is a subspace of $$V$$.

## Exercise $$\PageIndex{18}$$

Let $$M,N$$ be subspaces of a vector space $$V$$. Then $$M ∩N$$ consists of all vectors which are in both $$M$$ and $$N$$. Show that $$M ∩N$$ is a subspace of $$V$$.

## Exercise $$\PageIndex{19}$$

Let $$M,N$$ be subspaces of a vector space $$\mathbb{R}^2$$. Then $$N ∪M$$ consists of all vectors which are in either $$M$$ or $$N$$. Show that $$N ∪M$$ is not necessarily a subspace of $$\mathbb{R}^2$$ by giving an example where $$N ∪M$$ fails to be a subspace.

## Exercise $$\PageIndex{20}$$

Let $$X$$ consist of the real valued functions which are defined on an interval $$[a,b]$$. For $$f ,g ∈ X, f +g$$ is the name of the function which satisfies $$(f +g) (x) = f (x) +g(x)$$. For $$s$$ a real number, $$(s f) (x) = s(f (x))$$. Show this is a vector space.

The axioms of a vector space all hold because they hold for a vector space. The only thing left to verify is the assertions about the things which are supposed to exist. $$0$$ would be the zero function which sends everything to $$0$$. This is an additive identity. Now if $$f$$ is a function, $$−f (x) ≡ (−f (x))$$. Then $(f + (−f)) (x) ≡ f (x) + (−f) (x) ≡ f (x) + (−f (x)) = 0\nonumber$ Hence $$f + −f = 0$$. For each $$x ∈ [a,b]$$, let $$f_x (x) = 1$$ and $$f_x (y) = 0$$ if $$y\neq x$$. Then these vectors are obviously linearly independent.

## Exercise $$\PageIndex{21}$$

Consider functions defined on $$\{1, 2,\cdots ,n\}$$ having values in $$\mathbb{R}$$. Explain how, if $$V$$ is the set of all such functions, $$V$$ can be considered as $$\mathbb{R}^n$$.

Let $$f (i)$$ be the $$i$$th component of a vector $$\vec{x} ∈ \mathbb{R}^n$$. Thus a typical element in $$\mathbb{R}^n$$ is $$(f (1),\cdots , f (n))$$.

## Exercise $$\PageIndex{22}$$

Let the vectors be polynomials of degree no more than $$3$$. Show that with the usual definitions of scalar multiplication and addition wherein, for $$p(x)$$ a polynomial, $$(ap) (x) = ap(x)$$ and for $$p,q$$ polynomials $$(p+q) (x) = p(x) +q(x)$$, this is a vector space.

This is just a subspace of the vector space of functions because it is closed with respect to vector addition and scalar multiplication. Hence this is a vector space.

## Exercise $$\PageIndex{23}$$

Let $$V$$ be a vector space and suppose $$\{\vec{x}_1,\cdots ,\vec{x}_l\}$$ is a set of vectors in $$V$$. Show that $$\vec{0}$$ is in $$span\{\vec{x}_1,\cdots ,\vec{x}_k\}$$.

$$\sum\limits_{i=1}^k0\vec{x}_k=\vec{0}$$

## Exercise $$\PageIndex{24}$$

Determine if $$p(x) = 4x^2 −x$$ is in the span given by $span \{x^2+x,\:x^2-1,\:-x+2\}\nonumber$

## Exercise $$\PageIndex{25}$$

Determine if $$p(x) = −x^2 +x+2$$ is in the span given by $span\{ x^2 +x+1,\: 2x^2 +x\}\nonumber$

## Exercise $$\PageIndex{26}$$

Determine if $$A=\left[\begin{array}{cc}1&3\\0&0\end{array}\right]$$ is in the span given by $span\left\{\left[\begin{array}{cc}1&0\\0&1\end{array}\right],\:\left[\begin{array}{cc}0&1\\1&0\end{array}\right],\:\left[\begin{array}{cc}1&0\\1&1\end{array}\right],\:\left[\begin{array}{cc}0&1\\1&1\end{array}\right]\right\}\nonumber$

## Exercise $$\PageIndex{27}$$

Show that the spanning set in Exercise $$\PageIndex{26}$$ is a spanning set for $$M_{22}$$, the vector space of all $$2\times 2$$ matrices.

## Exercise $$\PageIndex{28}$$

Consider the vector space of polynomials of degree at most $$2$$, $$\mathbb{P}_2$$. Determine whether the following is a basis for $$\mathbb{P}_2$$. $\{x^2 +x+1,\: 2x^2 +2x+1,\: x+1\}\nonumber$ Hint: There is a isomorphism from $$\mathbb{R}^3$$ to $$\mathbb{P}_2$$. It is defined as follows: $T\vec{e}_1 = 1,\: T\vec{e}_2 = x,\: T\vec{e}_3= x^2\nonumber$ Then extend $$T$$ linearly. Thus $T\left[\begin{array}{c}1\\1\\1\end{array}\right]=x^2+x+1,\:T\left[\begin{array}{c}1\\2\\2\end{array}\right]=2x^2+2x+1,\:T\left[\begin{array}{c}1\\1\\0\end{array}\right]=1+x\nonumber$ It follows that if $\left\{\left[\begin{array}{c}1\\1\\1\end{array}\right],\:\left[\begin{array}{c}1\\2\\2\end{array}\right],\:\left[\begin{array}{c}1\\1\\0\end{array}\right]\right\}\nonumber$ is a basis for $$\mathbb{R}^3$$, then the polynomials will be a basis for $$\mathbb{P}_2$$ because they will be independent. Recall that an isomorphism takes a linearly independent set to a linearly independent set. Also, since $$T$$ is an isomorphism, it preserves all linear relations.

## Exercise $$\PageIndex{29}$$

Find a basis in $$\mathbb{P}_2$$ for the subspace $span\{ 1+x+x^2 ,\: 1+2x,\: 1+5x−3x^2\}\nonumber$ If the above three vectors do not yield a basis, exhibit one of them as a linear combination of the others. Hint: This is the situation in which you have a spanning set and you want to cut it down to form a linearly independent set which is also a spanning set. Use the same isomorphism above. Since $$T$$ is an isomorphism, it preserves all linear relations so if such can be found in $$\mathbb{R}^3$$, the same linear relations will be present in $$\mathbb{P}_2$$.

## Exercise $$\PageIndex{30}$$

Find a basis in $$\mathbb{P}_3$$ for the subspace $span\{ 1+x−x^2 +x^3 ,\: 1+2x+3x^3 ,\:−1+3x+5x^2 +7x^3 ,\: 1+6x+4x^2 +11x^3\}\nonumber$ If the above three vectors do not yield a basis, exhibit one of them as a linear combination of the others.

## Exercise $$\PageIndex{31}$$

Find a basis in $$\mathbb{P}_3$$ for the subspace $span\{ 1+x−x^2 +x^3 ,\: 1+2x+3x^3 ,\:−1+3x+5x^2 +7x^3 ,\: 1+6x+4x^2 +11x^3\}\nonumber$ If the above three vectors do not yield a basis, exhibit one of them as a linear combination of the others.

## Exercise $$\PageIndex{32}$$

Find a basis in $$\mathbb{P}_3$$ for the subspace $span\{ x^3 −2x^2 +x+2,\: 3x^3 −x^2 +2x+2,\: 7x^3 +x^2 +4x+2,\: 5x^3 +3x+2\}\nonumber$ If the above three vectors do not yield a basis, exhibit one of them as a linear combination of the others.

## Exercise $$\PageIndex{33}$$

Find a basis in $$\mathbb{P}_3$$ for the subspace $span\{ x^3 +2x^2 +x−2,\: 3x^3 +3x^2 +2x−2,\: 3x^3 +x+2,\: 3x^3 +x+2\}\nonumber$ If the above three vectors do not yield a basis, exhibit one of them as a linear combination of the others.

## Exercise $$\PageIndex{34}$$

Find a basis in $$\mathbb{P}_3$$ for the subspace $span\{ x^3 −5x^2 +x+5,\: 3x^3 −4x^2 +2x+5,\: 5x^3 +8x^2 +2x−5,\: 11x^3 +6x+5\}\nonumber$ If the above three vectors do not yield a basis, exhibit one of them as a linear combination of the others.

## Exercise $$\PageIndex{35}$$

Find a basis in $$\mathbb{P}_3$$ for the subspace $span\{x^3 −3x^2 +x+3,\: 3x^3 −2x^2 +2x+3,\: 7x^3 +7x^2 +3x−3,\: 7x^3 +4x+3\}\nonumber$ If the above three vectors do not yield a basis, exhibit one of them as a linear combination of the others.

## Exercise $$\PageIndex{36}$$

Find a basis in $$\mathbb{P}_3$$ for the subspace $span\{ x^3 −x^2 +x+1,\: 3x^3 +2x+1,\: 4x^3 +x^2 +2x+1,\: 3x^3 +2x−1\}\nonumber$ If the above three vectors do not yield a basis, exhibit one of them as a linear combination of the others.

## Exercise $$\PageIndex{37}$$

Find a basis in $$\mathbb{P}_3$$ for the subspace $span\{ x^3 −x^2 +x+1,\: 3x^3 +2x+1,\: 13x^3 +x^2 +8x+4,\: 3x^3 +2x−1\}\nonumber$ If the above three vectors do not yield a basis, exhibit one of them as a linear combination of the others.

## Exercise $$\PageIndex{38}$$

Find a basis in $$\mathbb{P}_3$$ for the subspace $span\{ x^3 −3x^2 +x+3,\: 3x^3 −2x^2 +2x+3,\:−5x^3 +5x^2 −4x−6,\: 7x^3 +4x−3\}\nonumber$ If the above three vectors do not yield a basis, exhibit one of them as a linear combination of the others.

## Exercise $$\PageIndex{39}$$

Find a basis in $$\mathbb{P}_3$$ for the subspace $span\{ x^3 −2x^2 +x+2,\: 3x^3 −x^2 +2x+2,\: 7x^3 −x^2 +4x+4,\: 5x^3 +3x−2\}\nonumber$ If the above three vectors do not yield a basis, exhibit one of them as a linear combination of the others.

## Exercise $$\PageIndex{40}$$

Find a basis in $$\mathbb{P}_3$$ for the subspace $span\{ x^3 −2x^2 +x+2,\: 3x^3 −x^2 +2x+2,\: 3x^3 +4x^2 +x−2,\: 7x^3 −x^2 +4x+4\}\nonumber$ If the above three vectors do not yield a basis, exhibit one of them as a linear combination of the others.

## Exercise $$\PageIndex{41}$$

Find a basis in $$\mathbb{P}_3$$ for the subspace $span\{ x^3 −4x^2 +x+4,\: 3x^3 −3x^2 +2x+4,\:−3x^3 +3x^2 −2x−4,\:−2x^3 +4x^2 −2x−4\}\nonumber$ If the above three vectors do not yield a basis, exhibit one of them as a linear combination of the others.

## Exercise $$\PageIndex{42}$$

Find a basis in $$\mathbb{P}_3$$ for the subspace $span\{ x^3 +2x^2 +x−2,\: 3x^3 +3x^2 +2x−2,\: 5x^3 +x^2 +2x+2,\: 10x^3 +10x^2 +6x−6\}\nonumber$ If the above three vectors do not yield a basis, exhibit one of them as a linear combination of the others.

## Exercise $$\PageIndex{43}$$

Find a basis in $$\mathbb{P}_3$$ for the subspace $span\{ x^3 +x^2 +x−1,\: 3x^3 +2x^2 +2x−1,\: x^3 +1,\: 4x^3 +3x^2 +2x−1\}\nonumber$ If the above three vectors do not yield a basis, exhibit one of them as a linear combination of the others.

## Exercise $$\PageIndex{44}$$

Find a basis in $$\mathbb{P}_3$$ for the subspace $span\{ x^3 −x^2 +x+1,\: 3x^3 +2x+1,\: x^3 +2x^2 −1,\: 4x^3 +x^2 +2x+1\}\nonumber$ If the above three vectors do not yield a basis, exhibit one of them as a linear combination of the others.

## Exercise $$\PageIndex{45}$$

Here are some vectors. $\{ x^3 +x^2 −x−1,\: 3x^3 +2x^2 +2x−1\}\nonumber$ If these are linearly independent, extend to a basis for all of $$\mathbb{P}_3$$.

## Exercise $$\PageIndex{46}$$

Here are some vectors. $\{ x^3 −2x^2 −x+2,\: 3x^3 −x^2 +2x+2\}\nonumber$ If these are linearly independent, extend to a basis for all of $$\mathbb{P}_3$$.

## Exercise $$\PageIndex{47}$$

Here are some vectors. $\{ x^3 −3x^2 −x+3,\: 3x^3 −2x^2 +2x+3\}\nonumber$ If these are linearly independent, extend to a basis for all of $$\mathbb{P}_3$$.

## Exercise $$\PageIndex{48}$$

Here are some vectors. $\{ x^3 −2x^2 −3x+2,\: 3x^3 −x^2 −6x+2,\:−8x^3 +18x+10\}\nonumber$ If these are linearly independent, extend to a basis for all of $$\mathbb{P}_3$$.

## Exercise $$\PageIndex{49}$$

Here are some vectors. $\{ x^3 −3x^2 −3x+3,\: 3x^3 −2x^2 −6x+3,\:−8x^3 +18x+40\}\nonumber$ If these are linearly independent, extend to a basis for all of $$\mathbb{P}_3$$.

## Exercise $$\PageIndex{50}$$

Here are some vectors. $\{ x^3 −x^2 +x+1,\: 3x^3 +2x+1,\: 4x^3 +2x+2\}\nonumber$ If these are linearly independent, extend to a basis for all of $$\mathbb{P}_3$$.

## Exercise $$\PageIndex{51}$$

Here are some vectors. $\{ x^3 +x^2 +2x−1,\: 3x^3 +2x^2 +4x−1,\: 7x^3 +8x+23\}\nonumber$ If these are linearly independent, extend to a basis for all of $$\mathbb{P}_3$$.

## Exercise $$\PageIndex{52}$$

Determine if the following set is linearly independent. If it is linearly dependent, write one vector as a linear combination of the other vectors in the set. $\{ x+1,\: x^2 +2,\: x^2 −x−3\}\nonumber$

## Exercise $$\PageIndex{53}$$

Determine if the following set is linearly independent. If it is linearly dependent, write one vector as a linear combination of the other vectors in the set. $\{ x^2 +x,\:−2x^2 −4x−6,\: 2x−2\}\nonumber$

## Exercise $$\PageIndex{54}$$

Determine if the following set is linearly independent. If it is linearly dependent, write one vector as a linear combination of the other vectors in the set. $\left\{\left[\begin{array}{cc}1&2\\0&1\end{array}\right],\:\left[\begin{array}{rr}-7&2\\-2&-3\end{array}\right],\:\left[\begin{array}{cc}4&0\\1&2\end{array}\right]\right\}\nonumber$

## Exercise $$\PageIndex{55}$$

Determine if the following set is linearly independent. If it is linearly dependent, write one vector as a linear combination of the other vectors in the set. $\left\{\left[\begin{array}{cc}1&0\\0&1\end{array}\right],\:\left[\begin{array}{cc}0&1\\0&1\end{array}\right],\:\left[\begin{array}{cc}1&0\\1&0\end{array}\right],\:\left[\begin{array}{cc}0&0\\1&1\end{array}\right]\right\}\nonumber$

## Exercise $$\PageIndex{56}$$

If you have $$5$$ vectors in $$\mathbb{R}^5$$ and the vectors are linearly independent, can it always be concluded they span $$\mathbb{R}^5$$?

Yes. If not, there would exist a vector not in the span. But then you could add in this vector and obtain a linearly independent set of vectors with more vectors than a basis.

## Exercise $$\PageIndex{57}$$

If you have $$6$$ vectors in $$\mathbb{R}^5$$, is it possible they are linearly independent? Explain.

No. They can't be.

## Exercise $$\PageIndex{58}$$

Let $$\mathbb{P}_3$$ be the polynomials of degree no more than $$3$$. Determine which of the following are bases for this vector space.

1. $$\{ x+1,\: x^3 +x^2 +2x,\: x^2 +x,\: x^3 +x^2 +x\}$$
2. $$\{ x^3 +1,\: x^2 +x,\: 2x^3 +x^2 ,\: 2x^3 −x^2 −3x+1\}$$
1. Suppose $c_1(x^3 +1)+c_2 (x^2 +x) +c_3( 2x^3 +x^2) +c_4 (2x^3 −x^2 −3x+1) = 0\nonumber$ Then combine the terms according to power of $$x$$. $(c_1 +2c_3 +2c_4) x^3 + (c_2 +c_3 −c_4) x^2 + (c_2 −3c_4) x+ (c_1 +c_4) = 0\nonumber$ Is there a non zero solution to the system \begin{aligned}c_1 +2c_3 +2c_4 &= 0\\ c_2 +c_3 −c_4 &= 0\\ c_2 −3c_4 &= 0\\ c_1 +c_4 &= 0\end{aligned}, Solution is: $[c_1 = 0,\: c_2 = 0,\: c_3 = 0,\: c_4 = 0]\nonumber$ Therefore, these are linearly independent.

## Exercise $$\PageIndex{59}$$

In the context of the above problem, consider polynomials $\{a_ix^3 +b_ix^2 +c_ix+d_i ,\: i = 1, 2, 3, 4\}\nonumber$ Show that this collection of polynomials is linearly independent on an interval $$[s,t]$$ if and only if $\left[\begin{array}{cccc}a_1&b_1&c_1&d_1 \\ a_2&b_2&c_2&d_2 \\ a_3&b_3&c_3&d_3\\ a_4&b_4&c_4&d_4\end{array}\right]\nonumber$ is an invertible matrix.

Let $$p_i(x)$$ denote the $$i$$th of these polynomials. Suppose $$\sum_i C_ip_i(x) = 0$$. Then collecting terms according to the exponent of $$x$$, you need to have \begin{aligned}C_1a_1 +C_2a_2 +C_3a_3 +C_4a_4 &= 0\\ C_1b_1 +C_2b_2 +C_3b_3 +C_4b_4 &= 0\\ C_1c_1 +C_2c_2 +C_3c_3 +C_4c_4 &= 0\\ C_1d_1 +C_2d_2 +C_3d_3 +C_4d_4 &= 0\end{aligned} The matrix of coefficients is just the transpose of the above matrix. There exists a non trivial solution if and only if the determinant of this matrix equals $$0$$.

## Exercise $$\PageIndex{60}$$

Let the field of scalars be $$\mathbb{Q}$$, the rational numbers and let the vectors be of the form $$a+b\sqrt{2}$$ where $$a,b$$ are rational numbers. Show that this collection of vectors is a vector space with field of scalars $$\mathbb{Q}$$ and give a basis for this vector space.

When you add two of these you get one and when you multiply one of these by a scalar, you get another one. A basis is $$\{1,\sqrt{2}\}$$. By definition, the span of these gives the collection of vectors. Are they independent? Say $$a + b\sqrt{2} = 0$$ where $$a,b$$ are rational numbers. If $$a\neq 0$$, then $$b\sqrt{2} = −a$$ which can’t happen since a is rational. If $$b\neq 0$$, then $$−a = b\sqrt{2}$$ which again can’t happen because on the left is a rational number and on the right is an irrational. Hence both $$a,b = 0$$ and so this is a basis.

## Exercise $$\PageIndex{61}$$

Suppose $$V$$ is a finite dimensional vector space. Based on the exchange theorem above, it was shown that any two bases have the same number of vectors in them. Give a different proof of this fact using the earlier material in the book. Hint: Suppose $$\{\vec{x}_1,\cdots ,\vec{x}_n\}$$ and $$\{\vec{y}_1,\cdots , \vec{y}_m\}$$ are two bases with $$m < n$$. Then define $φ : \mathbb{R}^n \mapsto V,\: ψ :\mathbb{R}^m\mapsto V\nonumber$ by $φ (\vec{a}) = \sum\limits_{k=1}^n a_k\vec{x}_k ,\: ψ(\vec{b}) =\sum\limits_{j=1}^m b_j\vec{y}_j\nonumber$ Consider the linear transformation, $$ψ^{−1}\circ φ$$. Argue it is a one to one and onto mapping from $$\mathbb{R}^n$$ to $$\mathbb{R}^m$$. Now consider a matrix of this linear transformation and its reduced row-echelon form.

This is obvious because when you add two of these you get one and when you multiply one of these by a scalar, you get another one. A basis is $$\{1,\sqrt{2}\}$$. By definition, the span of these gives the collection of vectors. Are they independent? Say $$a+b\sqrt{2} = 0$$ where $$a,b$$ are rational numbers. If $$a\neq 0$$, then $$b\sqrt{2} = −a$$ which can’t happen since $$a$$ is rational. If $$b\neq 0$$, then $$−a = b\sqrt{2}$$ which again can’t happen because on the left is a rational number and on the right is an irrational. Hence both $$a,b = 0$$ and so this is a basis.

## Exercise $$\PageIndex{62}$$

Let $$M =\{\vec{u} = (u_1,\:u_2,\:u_3,\:u_4)\in \mathbb{R}^4\: :\: |u_1| ≤ 4\}$$. Is $$M$$ a subspace of $$\mathbb{R}^4$$?

This is not a subspace. $$\left[\begin{array}{c}1\\1\\1\\1\end{array}\right]$$ is in it, but $$20\left[\begin{array}{c}1\\1\\1\\1\end{array}\right]$$ is not.

## Exercise $$\PageIndex{63}$$

Let $$M =\{\vec{u} = (u_1,\:u_2,\:u_3,\:u_4)\in \mathbb{R}^4\: :\: \sin(u_1) = 1\}$$. Is $$M$$ a subspace of $$\mathbb{R}^4$$?

This is not a subspace.

## Exercise $$\PageIndex{64}$$

Let $$W$$ be a subset of $$M_{22}$$ given by $W = \{ A|A\in M_{22},A^T = A\}\nonumber$ In words, $$W$$ is the set of all symmetric $$2\times 2$$ matrices. Is $$W$$ a subspace of $$M_{22}$$?

## Exercise $$\PageIndex{65}$$

Let $$W$$ be a subset of $$M_{22}$$ given by $W=\left\{\left[\begin{array}{cc}a&b\\c&d\end{array}\right] \: |a,b,c,d\in\mathbb{R},\:a+b=c+d\right\}\nonumber$ Is $$W$$ a subspace of $$M_{22}$$?

## Exercise $$\PageIndex{66}$$

Let $$W$$ be a subset of $$P_3$$ given by $W = \{ ax^3 +bx^2 +cx+d|\: a,b, c,d\in\mathbb{R},d = 0\}\nonumber$ Is $$W$$ a subspace of $$P_3$$?

## Exercise $$\PageIndex{67}$$

Let $$W$$ be a subset of $$P_3$$ given by $W = \{ p(x) = ax^3 +bx^2 +cx+d|\: a,b, c,d\in\mathbb{R}, p(2) = 1\}\nonumber$ Is $$W$$ a subspace of $$P_3$$?

## Exercise $$\PageIndex{68}$$

Let $$T$$: $$\mathbb{P}_2\to\mathbb{R}$$ be a linear transformation such that $T(x^2)=1;\: T(x^2+x)=5;\: T(x^2+x+1)=-1.\nonumber$ Find $$T(ax^2+bx+c)$$.

By linearity we have $$T(x^2 ) = 1,\: T(x) = T(x^2 +x−x^2 ) = T(x^2 +x)−T (x^2 ) = 5−1 = 5,$$ and $$T(1) = T(x^2 +x+1−(x^2 +x)) = T(x^2 +x+1)−T(x^2 +x)) = −1−5 = −6$$. Thus $$T(ax^2 +bx+c) = aT(x^2 ) +bT(x) +cT(1) = a+5b−6c$$.

## Exercise $$\PageIndex{69}$$

Consider the following functions $$T$$: $$\mathbb{R}^3\to\mathbb{R}^2$$. Explain why each of these functions $$T$$ is not linear.

1. $$T\left[\begin{array}{c}x\\y\\z\end{array}\right]=\left[\begin{array}{c}x+2y+3z+1 \\ 2y-3x+z\end{array}\right]$$
2. $$T\left[\begin{array}{c}x\\y\\z\end{array}\right]=\left[\begin{array}{c}x+2y^2+3z \\ 2y+3z+z\end{array}\right]$$
3. $$T\left[\begin{array}{c}x\\y\\z\end{array}\right]=\left[\begin{array}{c}\sin x+2y+3z \\ 2y+3z+z\end{array}\right]$$
4. $$T\left[\begin{array}{c}x\\y\\z\end{array}\right]=\left[\begin{array}{c}x+2y+3z \\ 2y+3z-\ln z\end{array}\right]$$

## Exercise $$\PageIndex{70}$$

Suppose $$T$$ is a linear transformation such that \begin{aligned} T\left[\begin{array}{r}1\\1\\-7\end{array}\right]&=\left[\begin{array}{c}3\\3\\3\end{array}\right] \\ T\left[\begin{array}{r}-1\\0\\6\end{array}\right]&=\left[\begin{array}{c}1\\2\\3\end{array}\right] \\ T\left[\begin{array}{r}0\\-1\\2\end{array}\right]&=\left[\begin{array}{r}1\\3\\-1\end{array}\right]\end{aligned} Find the matrix of $$T$$. That is find $$A$$ such that $$T(\vec{x})=A\vec{x}$$.

$\left[\begin{array}{rrr}3&1&1\\3&2&3\\3&3&-1\end{array}\right]\left[\begin{array}{ccc}6&2&1\\5&2&1\\6&1&1\end{array}\right]=\left[\begin{array}{ccc}29&9&5\\46&13&8\\27&11&5\end{array}\right]\nonumber$

## Exercise $$\PageIndex{71}$$

Suppose $$T$$ is a linear transformation such that \begin{aligned} T\left[\begin{array}{r}1\\2\\-18\end{array}\right]&=\left[\begin{array}{c}5\\2\\5\end{array}\right] \\ T\left[\begin{array}{r}-1\\-1\\15\end{array}\right]&=\left[\begin{array}{c}3\\3\\5\end{array}\right] \\ T\left[\begin{array}{r}0\\-1\\4\end{array}\right]&=\left[\begin{array}{r}2\\5\\-2\end{array}\right]\end{aligned} Find the matrix of $$T$$. That is find $$A$$ such that $$T(\vec{x})=A\vec{x}$$.

$\left[\begin{array}{rrr}5&3&2\\2&3&5\\5&5&-2\end{array}\right]\left[\begin{array}{ccc}11&4&1\\10&4&1\\12&3&1\end{array}\right]=\left[\begin{array}{ccc}109&38&10\\112&35&10\\81&34&8\end{array}\right]\nonumber$

## Exercise $$\PageIndex{72}$$

Consider the following functions $$T$$: $$\mathbb{R}^3\to\mathbb{R}^2$$. Show that each is a linear transformation and determine for each the matrix $$A$$ such that $$T(\vec{x}) = A\vec{x}$$.

1. $$T\left[\begin{array}{c}x\\y\\z\end{array}\right]=\left[\begin{array}{c}x+2y+3z \\ 2y-3x+z\end{array}\right]$$
2. $$T\left[\begin{array}{c}x\\y\\z\end{array}\right]=\left[\begin{array}{c}7x+2y+z \\ 3x-11y+2z\end{array}\right]$$
3. $$T\left[\begin{array}{c}x\\y\\z\end{array}\right]=\left[\begin{array}{c}3x+2y+z \\ x+2y+6z\end{array}\right]$$
4. $$T\left[\begin{array}{c}x\\y\\z\end{array}\right]=\left[\begin{array}{c}2y-5x+z \\ x+y+z\end{array}\right]$$

## Exercise $$\PageIndex{73}$$

Suppose $[A_1\cdots A_n]^{-1}\nonumber$ exists where each $$A_j\in\mathbb{R}^n$$ and let vectors $$\{B_1,\cdots ,B_n\}$$ in $$\mathbb{R}^m$$ be given. Show that there always exists a linear transformation $$T$$ such that $$T(A_i)=B_i$$.

## Exercise $$\PageIndex{74}$$

Let $$V$$ and $$W$$ be subspaces of $$\mathbb{R}^n$$ and $$\mathbb{R}^m$$ respectively and let $$T$$: $$V → W$$ be a linear transformation. Suppose that $$\{T\vec{v}_1,\cdots ,T\vec{v}_r\}$$ is linearly independent. Show that it must be the case that $$\{\vec{v}_1,\cdots ,\vec{v}_r\}$$ is also linearly independent.

If $$\sum\limits_i^ra_i\vec{v}_r=0$$, then using linearity properties of $$T$$ we get $0=T(0)=T\left(\sum\limits_i^ra_i\vec{v}_r\right)=\sum\limits_i^ra_iT(\vec{v}_r).\nonumber$ Since we assume that $$\{T\vec{v}_1,\cdots ,T\vec{v}_r\}$$ is linearly independent, we must have all $$a_i = 0$$, and therefore we conclude that $$\{\vec{v}_1,\cdots ,\vec{v}_r\}$$ is also linearly independent.

## Exercise $$\PageIndex{75}$$

Let $V=span\left\{\left[\begin{array}{c}1\\1\\2\\0\end{array}\right],\:\left[\begin{array}{c}0\\1\\1\\1\end{array}\right],\:\left[\begin{array}{c}1\\1\\0\\1\end{array}\right]\right\}\nonumber$ Let $$T\vec{x}=A\vec{x}$$ where $$A$$ is the matrix $\left[\begin{array}{cccc}1&1&1&1\\0&1&1&0\\0&1&2&1\\1&1&1&2\end{array}\right]\nonumber$ Give a basis for $$Im(T)$$.

## Exercise $$\PageIndex{76}$$

Let $V=span\left\{\left[\begin{array}{c}1\\0\\0\\1\end{array}\right],\:\left[\begin{array}{c}1\\1\\1\\1\end{array}\right],\:\left[\begin{array}{c}1\\4\\4\\1\end{array}\right]\right\}\nonumber$ Let $$T\vec{x}=A\vec{x}$$ where $$A$$ is the matrix $\left[\begin{array}{cccc}1&1&1&1\\0&1&1&0\\0&1&2&1\\1&1&1&2\end{array}\right]\nonumber$ Find a basis for $$Im(T)$$. In this case, the original vectors do not form an independent set.

Since the third vector is a linear combinations of the first two, then the image of the third vector will also be a linear combinations of the image of the first two. However the image of the first two vectors are linearly independent (check!), and hence form a basis of the image. Thus a basis for $$Im(T)$$ is: $V=span\left\{\left[\begin{array}{c}2\\0\\1\\3\end{array}\right],\:\left[\begin{array}{c}4\\2\\4\\5\end{array}\right]\right\}\nonumber$

## Exercise $$\PageIndex{77}$$

If $$\{\vec{v}_1,\cdots ,\vec{v}_r\}$$ is linearly independent and $$T$$ is a one to one linear transformation, show that $$\{T\vec{v}_1,\cdots ,T\vec{v}_r\}$$ is also linearly independent. Give an example which shows that if $$T$$ is only linear, it can happen that, although $$\{\vec{v}_1,\cdots ,\vec{v}_r\}$$ is linearly independent, $$\{T\vec{v}_1,\cdots ,T\vec{v}_r\}$$ is not. In fact, show that it can happen that each of the $$T\vec{v}_j$$ equals $$0$$.

## Exercise $$\PageIndex{78}$$

Let $$V$$ and $$W$$ be subspaces of $$\mathbb{R}^n$$ and $$\mathbb{R}^m$$ respectively and let $$T$$: $$V → W$$ be a linear transformation. Show that if $$T$$ is onto $$W$$ and if $$\{\vec{v}_1,\cdots ,\vec{v}_r\}$$ is a basis for $$V$$, then $$span\{T\vec{v}_1,\cdots ,T\vec{v}_r\} = W$$.

## Exercise $$\PageIndex{79}$$

Define $$T$$: $$\mathbb{R}^4 → \mathbb{R}^3$$ as follows. $T\vec{x}=\left[\begin{array}{rrrr}3&2&1&8\\2&2&-2&6\\1&1&-1&3\end{array}\right]\vec{x}\nonumber$ Find a basis for $$Im(T)$$. Also find a basis for $$\text{ker}(T)$$.

## Exercise $$\PageIndex{80}$$

Define $$T$$: $$\mathbb{R}^4 → \mathbb{R}^3$$ as follows. $T\vec{x}=\left[\begin{array}{rrr}1&2&0\\1&1&1\\0&1&1\end{array}\right]\vec{x}\nonumber$ where on the right, it is just matrix multiplication of the vector $$\vec{x}$$ which is meant. Explain why $$T$$ is an isomorphism of $$\mathbb{R}^3$$ to $$\mathbb{R}^3$$.

## Exercise $$\PageIndex{81}$$

Suppose $$T$$: $$\mathbb{R}^3 → \mathbb{R}^3$$ is a linear transformation given by $T\vec{x}=A\vec{x}\nonumber$ where $$A$$ is a $$3\times 3$$ matrix. Show that $$T$$ is an isomorphism if and only if $$A$$ is invertible.

## Exercise $$\PageIndex{82}$$

Suppose $$T$$: $$\mathbb{R}^3 → \mathbb{R}^3$$ is a linear transformation given by $T\vec{x}=A\vec{x}\nonumber$ where $$A$$ is a $$m\times n$$ matrix. Show that $$T$$ is never an isomorphism if $$m\neq n$$. In particular, show that if $$m>n$$, $$T$$ cannot be onto and if $$m<n$$, then $$T$$ cannot be one to one.

## Exercise $$\PageIndex{83}$$

Define $$T$$: $$\mathbb{R}^2 → \mathbb{R}^3$$ as follows. $T\vec{x}=\left[\begin{array}{cc}1&0\\1&1\\0&1\end{array}\right]\vec{x}\nonumber$ where on the right, it is just matrix multiplication of the vector $$\vec{x}$$ which is meant. Show that $$T$$ is one to one. Next let $$W = Im(T)$$. Show that $$T$$ is an isomorphism of $$\mathbb{R}^2$$ and $$Im (T)$$.

## Exercise $$\PageIndex{84}$$

In the above problem, find a $$2\times 3$$ matrix $$A$$ such that the restriction of $$A$$ to $$Im(T)$$ gives the same result as $$T^{−1}$$ on $$Im(T)$$. Hint: You might let $$A$$ be such that $A\left[\begin{array}{c}1\\1\\0\end{array}\right]=\left[\begin{array}{c}1\\0\end{array}\right],\:A\left[\begin{array}{c}0\\1\\1\end{array}\right]=\left[\begin{array}{c}0\\1\end{array}\right]\nonumber$ now find another vector $$\vec{v} ∈ \mathbb{R}^3$$ such that $\left\{\left[\begin{array}{c}1\\1\\0\end{array}\right],\:\left[\begin{array}{c}0\\1\\1\end{array}\right],\:\vec{v}\right\}\nonumber$ is a basis. You could pick $\vec{v}=\left[\begin{array}{c}0\\0\\1\end{array}\right]\nonumber$ for example. Explain why this one works or one of your choice works. Then you could define $$A\vec{v}$$ to equal some vector in $$\mathbb{R}^2$$. Explain why there will be more than one such matrix $$A$$ which will deliver the inverse isomorphism $$T^{−1}$$ on $$Im(T)$$.

## Exercise $$\PageIndex{85}$$

Now let $$V$$ equal $$span\left\{\left[\begin{array}{c}1\\0\\1\end{array}\right],\:\left[\begin{array}{c}0\\1\\1\end{array}\right]\right\}$$ and let $$T$$: $$V\to W$$ be a linear transformation where $W=span\left\{\left[\begin{array}{c}1\\0\\1\\0\end{array}\right],\:\left[\begin{array}{c}0\\1\\1\\1\end{array}\right]\right\}\nonumber$ and $T\left[\begin{array}{c}1\\0\\1\end{array}\right]=\left[\begin{array}{c}1\\0\\1\\0\end{array}\right],\:T\left[\begin{array}{c}0\\1\\1\end{array}\right]=\left[\begin{array}{c}0\\1\\1\\1\end{array}\right]\nonumber$

Explain why $$T$$ is an isomorphism. Determine a matrix $$A$$ which, when multiplied on the left gives the same result as $$T$$ on $$V$$ and a matrix $$B$$ which delivers $$T^{−1}$$ on $$W$$. Hint: You need to have $A\left[\begin{array}{cc}1&0\\0&1\\1&1\end{array}\right]=\left[\begin{array}{cc}1&0\\0&1\\1&1\\0&1\end{array}\right]\nonumber$

Now enlarge $$\left[\begin{array}{c}1\\0\\1\end{array}\right]$$, $$\left[\begin{array}{c}0\\1\\1\end{array}\right]$$ to obtain a basis for $$\mathbb{R}^3$$. You could add in $$\left[\begin{array}{c}0\\0\\1\end{array}\right]$$ for example, and then pick another vector in $$\mathbb{R}^4$$ and let $$A\left[\begin{array}{c}0\\0\\1\end{array}\right]$$ equal this other vector. Then you would have $A\left[\begin{array}{ccc}1&0&0\\0&1&0\\1&1&1\end{array}\right]=\left[\begin{array}{ccc}1&0&0\\0&1&0\\1&1&0\\0&1&1\end{array}\right]\nonumber$

This would involve picking for the new vector in $$\mathbb{R}^4$$ the vector $$\left[\begin{array}{cccc}0&0&0&1\end{array}\right]^T$$. Then you could find $$A$$. You can do something similar to find a matrix for $$T^{-1}$$ denoted as $$B$$.

## Exercise $$\PageIndex{86}$$

Let $$V=\mathbb{R}^3$$ and let $W=span(S),\text{ where }S=\left\{\left[\begin{array}{r}1\\-1\\1\end{array}\right],\:\left[\begin{array}{r}-2\\2\\-2\end{array}\right],\:\left[\begin{array}{r}-1\\1\\1\end{array}\right],\:\left[\begin{array}{r}1\\-1\\3\end{array}\right]\right\}\nonumber$ Find a basis of $$W$$ consisting of vectors in $$S$$.

In this case $$\text{dim}(W) = 1$$ and a basis for $$W$$ consisting of vectors in $$S$$ can be obtained by taking any (nonzero) vector from $$S$$.

## Exercise $$\PageIndex{87}$$

Let $$T$$ be a linear transformation given by $T\left[\begin{array}{c}x\\y\end{array}\right]=\left[\begin{array}{cc}1&1\\1&1\end{array}\right]\left[\begin{array}{c}x\\y\end{array}\right]\nonumber$ Find a basis for $$\text{ker}(T)$$ and $$Im(T)$$.

A basis for $$\text{ker}(T)$$ is $$\left\{\left[\begin{array}{r}1\\-1\end{array}\right]\right\}$$ and a basis for $$Im(T)$$ is $$\left\{\left[\begin{array}{r}1\\1\end{array}\right]\right\}$$. There are many other possibilities for the specific bases, but in this case $$\text{dim}(\text{ker}(T)) = 1$$ and $$\text{dim}(Im(T)) = 1$$.

## Exercise $$\PageIndex{88}$$

Let $$T$$ be a linear transformation given by $T\left[\begin{array}{c}x\\y\end{array}\right]=\left[\begin{array}{cc}1&0\\1&1\end{array}\right]\left[\begin{array}{c}x\\y\end{array}\right]\nonumber$ Find a basis for $$\text{ker}(T)$$ and $$Im(T)$$.

In this case $$\text{ker}(T) = \{0\}$$ and $$Im(T) = \mathbb{R}^2$$ (pick any basis of $$\mathbb{R}^2$$).

## Exercise $$\PageIndex{89}$$

Let $$V=\mathbb{R}^3$$ and let $W=span\left\{\left[\begin{array}{c}1\\1\\1\end{array}\right],\:\left[\begin{array}{r}-1\\2\\-1\end{array}\right]\right\}\nonumber$ Extend this basis of $$W$$ to a basis of $$V$$.

There are many possible such extensions, one is (how do we know?): $\left\{\left[\begin{array}{r}1\\1\\1\end{array}\right],\:\left[\begin{array}{r}-1\\2\\-1\end{array}\right],\:\left[\begin{array}{c}0\\0\\1\end{array}\right]\right\}\nonumber$

## Exercise $$\PageIndex{90}$$

Let $$T$$ be a linear transformation given by $T\left[\begin{array}{c}x\\y\\z\end{array}\right]=\left[\begin{array}{ccc}1&1&1\\1&1&1\end{array}\right]\left[\begin{array}{c}x\\y\\z\end{array}\right]\nonumber$ What is $$\text{dim}(\text{ker}(T))$$?

We can easily see that $$\text{dim}(Im(T)) = 1$$, and thus $$\text{dim}(\text{ker}(T)) = 3−\text{dim}(Im(T)) = 3−1 = 2$$.

## Exercise $$\PageIndex{91}$$

Consider the following functions which map $$\mathbb{R}^n$$ to $$\mathbb{R}^n$$.

1. $$T$$ multiplies the $$j$$th component of $$\vec{x}$$ by a nonzero number $$b$$.
2. $$T$$ replaces the $$i$$th component of $$\vec{x}$$ with $$b$$ times the $$j$$th component added to the $$i$$h component.
3. $$T$$ switches the $$i$$th and $$j$$th components.

Show these functions are linear transformations and describe their matrices $$A$$ such that $$T (\vec{x}) = A\vec{x}$$.

1. The matrix of $$T$$ is the elementary matrix which multiplies the $$j$$th diagonal entry of the identity matrix by $$b$$.
2. The matrix of $$T$$ is the elementary matrix which takes $$b$$ times the $$j$$th row and adds to the $$i$$th row.
3. The matrix of $$T$$ is the elementary matrix which switches the $$i$$th and the $$j$$th rows where the two components are in the $$i$$th and $$j$$th positions.

## Exercise $$\PageIndex{92}$$

You are given a linear transformation $$T$$: $$\mathbb{R}^n → \mathbb{R}^m$$ and you know that $T(A_i)=B_i\nonumber$ where $$\left[\begin{array}{ccc}A_1&\cdots&A_n\end{array}\right]^{-1}$$ exists. Show that the matrix of $$T$$ is of the form $\left[\begin{array}{ccc}B_1&\cdots&B_n\end{array}\right]\:\left[\begin{array}{ccc}A_1&\cdots&A_n\end{array}\right]^{-1}\nonumber$

Suppose $\left[\begin{array}{c}\vec{c}_1^T \\ \vdots \\ \vec{c}_n^T\end{array}\right]=\left[\begin{array}{ccc}\vec{a}_1&\cdots&\vec{a}_n\end{array}\right]^{-1}\nonumber$ Thus $$\vec{c}_i^T\vec{a}_j=\delta_{ij}$$. Therefore \begin{aligned} \left[\begin{array}{ccc}\vec{b}_1&\cdots&\vec{b}_n\end{array}\right]\: \left[\begin{array}{ccc}\vec{a}_1&\cdots&\vec{a}_n\end{array}\right]^{-1}\vec{a}_i &=\left[\begin{array}{ccc}\vec{b}_1&\cdots&\vec{b}_n\end{array}\right]\:\left[\begin{array}{c}\vec{c}_1^T \\ \vdots \\ \vec{c}_n^T\end{array}\right] \vec{a}_i \\ &=\left[\begin{array}{ccc}\vec{b}_1&\cdots&\vec{b}_n\end{array}\right] \vec{e}_i \\ &=\vec{b}_i\end{aligned} Thus $$T\vec{a}_i=\left[\begin{array}{ccc}\vec{b}_1&\cdots&\vec{b}_n\end{array}\right]\: \left[\begin{array}{ccc}\vec{a}_1&\cdots&\vec{a}_n\end{array}\right]^{-1}\vec{a}_i=A\vec{a}_i$$. If $$\vec{x}$$ is arbitrary, then since the matrix $$\left[\begin{array}{ccc}\vec{a}_1&\cdots&\vec{a}_n\end{array}\right]$$ is invertible, there exists a unique $$\vec{y}$$ such that $$\left[\begin{array}{ccc}\vec{a}_1&\cdots&\vec{a}_n\end{array}\right]\vec{y}=\vec{x}$$ Hence $T\vec{x}=T\left(\sum\limits_{i=1}^ny_i\vec{a}_i\right)=\sum\limits_{i=1}^ny_iT\vec{a}_i=\sum\limits_{i=1}^ny_1A\vec{a}_i=A\left(\sum\limits_{i=1}^ny_i\vec{a}_i\right)=A\vec{x}\nonumber$

## Exercise $$\PageIndex{93}$$

Suppose $$T$$ is a linear transformation such that \begin{aligned}T\left[\begin{array}{r}1\\2\\-6\end{array}\right]&=\left[\begin{array}{c}5\\1\\3\end{array}\right] \\ T\left[\begin{array}{r}-1\\-1\\5\end{array}\right]&=\left[\begin{array}{c}1\\1\\5\end{array}\right] \\ T\left[\begin{array}{r}0\\-1\\2\end{array}\right]&=\left[\begin{array}{r}5\\3\\-2\end{array}\right]\end{aligned} Find the matrix of $$T$$. That is find $$A$$ such that $$T(\vec{x}) = A\vec{x}$$.

$\left[\begin{array}{rrr}5&1&5\\1&1&3\\3&5&-2\end{array}\right]\left[\begin{array}{ccc}3&2&1\\2&2&1\\4&1&1\end{array}\right]=\left[\begin{array}{ccc}37&17&11\\17&7&5\\11&14&6\end{array}\right]\nonumber$

## Exercise $$\PageIndex{94}$$

Suppose $$T$$ is a linear transformation such that \begin{aligned}T\left[\begin{array}{r}1\\1\\-8\end{array}\right]&=\left[\begin{array}{c}1\\3\\1\end{array}\right] \\ T\left[\begin{array}{r}-1\\0\\6\end{array}\right]&=\left[\begin{array}{c}2\\4\\1\end{array}\right] \\ T\left[\begin{array}{r}0\\-1\\3\end{array}\right]&=\left[\begin{array}{r}6\\1\\-1\end{array}\right]\end{aligned} Find the matrix of $$T$$. That is find $$A$$ such that $$T(\vec{x}) = A\vec{x}$$.

$\left[\begin{array}{rrr}1&2&6\\3&4&1\\1&1&-1\end{array}\right]\left[\begin{array}{ccc}6&3&1\\5&3&1\\6&2&1\end{array}\right]=\left[\begin{array}{ccc}52&21&9\\44&23&8\\5&4&1\end{array}\right]\nonumber$

## Exercise $$\PageIndex{95}$$

Suppose $$T$$ is a linear transformation such that \begin{aligned}T\left[\begin{array}{r}1\\3\\-7\end{array}\right]&=\left[\begin{array}{c}-3\\1\\3\end{array}\right] \\ T\left[\begin{array}{r}-1\\-2\\6\end{array}\right]&=\left[\begin{array}{4}1\\3\\-3\end{array}\right] \\ T\left[\begin{array}{r}0\\-1\\2\end{array}\right]&=\left[\begin{array}{r}5\\3\\-3\end{array}\right]\end{aligned} Find the matrix of $$T$$. That is find $$A$$ such that $$T(\vec{x}) = A\vec{x}$$.

$\left[\begin{array}{rrr}-3&1&5\\1&3&3\\3&-3&-3\end{array}\right]\left[\begin{array}{ccc}2&2&1\\1&2&1\\4&1&1\end{array}\right]=\left[\begin{array}{rrr}15&1&3\\17&11&7\\-9&-3&-3\end{array}\right]\nonumber$

## Exercise $$\PageIndex{96}$$

Suppose $$T$$ is a linear transformation such that \begin{aligned}T\left[\begin{array}{r}1\\1\\-7\end{array}\right]&=\left[\begin{array}{c}3\\3\\3\end{array}\right] \\ T\left[\begin{array}{r}-1\\0\\6\end{array}\right]&=\left[\begin{array}{c}1\\2\\3\end{array}\right] \\ T\left[\begin{array}{r}0\\-1\\2\end{array}\right]&=\left[\begin{array}{r}1\\3\\-1\end{array}\right]\end{aligned} Find the matrix of $$T$$. That is find $$A$$ such that $$T(\vec{x}) = A\vec{x}$$.

$\left[\begin{array}{rrr}3&1&1\\3&2&3\\3&3&-1\end{array}\right]\left[\begin{array}{ccc}6&2&1\\5&2&1\\6&1&1\end{array}\right]=\left[\begin{array}{ccc}29&9&5\\46&13&8\\27&11&5\end{array}\right]\nonumber$

## Exercise $$\PageIndex{97}$$

Suppose $$T$$ is a linear transformation such that \begin{aligned}T\left[\begin{array}{r}1\\2\\-18\end{array}\right]&=\left[\begin{array}{c}5\\2\\5\end{array}\right] \\ T\left[\begin{array}{r}-1\\-1\\15\end{array}\right]&=\left[\begin{array}{c}3\\3\\5\end{array}\right] \\ T\left[\begin{array}{r}0\\-1\\4\end{array}\right]&=\left[\begin{array}{r}2\\5\\-2\end{array}\right]\end{aligned} Find the matrix of $$T$$. That is find $$A$$ such that $$T(\vec{x}) = A\vec{x}$$.

$\left[\begin{array}{rrr}5&3&2\\2&3&5\\5&5&-2\end{array}\right]\left[\begin{array}{ccc}11&4&1\\10&4&1\\12&3&1\end{array}\right]=\left[\begin{array}{ccc}109&38&10 \\112&35&10\\81&34&8\end{array}\right]\nonumber$

## Exercise $$\PageIndex{98}$$

Consider the following functions $$T$$: $$\mathbb{R}^3 → \mathbb{R}^2$$. Show that each is a linear transformation and determine for each the matrix $$A$$ such that $$T(\vec{x}) = A\vec{x}$$.

1. $$T\left[\begin{array}{c}x\\y\\z\end{array}\right]=\left[\begin{array}{c}x+2y+3z \\ 2y-3x+z\end{array}\right]$$
2. $$T\left[\begin{array}{c}x\\y\\z\end{array}\right]=\left[\begin{array}{c}7x+2y+z \\ 3x-11y+2z\end{array}\right]$$
3. $$T\left[\begin{array}{c}x\\y\\z\end{array}\right]=\left[\begin{array}{c}3x+2y+z \\ x+2y+6z\end{array}\right]$$
4. $$T\left[\begin{array}{c}x\\y\\z\end{array}\right]=\left[\begin{array}{c}2y-5x+z \\ x+y+z\end{array}\right]$$

## Exercise $$\PageIndex{99}$$

Consider the following functions $$T$$: $$\mathbb{R}^3 → \mathbb{R}^2$$. Explain why each of these functions $$T$$ is not linear.

1. $$T\left[\begin{array}{c}x\\y\\z\end{array}\right]=\left[\begin{array}{c}x+2y+3z+1 \\ 2y-3x+z\end{array}\right]$$
2. $$T\left[\begin{array}{c}x\\y\\z\end{array}\right]=\left[\begin{array}{c}x+2y^2+3z \\ 2y+3x+z\end{array}\right]$$
3. $$T\left[\begin{array}{c}x\\y\\z\end{array}\right]=\left[\begin{array}{c}\sin x+2y+3z \\ 2y+3x+z\end{array}\right]$$
4. $$T\left[\begin{array}{c}x\\y\\z\end{array}\right]=\left[\begin{array}{c}x+2y+3z \\ 2y+3x-\ln z\end{array}\right]$$

## Exercise $$\PageIndex{100}$$

Suppose $\left[\begin{array}{ccc}A_1&\cdots&A_n\end{array}\right]^{-1}\nonumber$ exists where each $$A_j ∈ \mathbb{R}^n$$ and let vectors $$\{B_1,\cdots ,B_n\}$$ in $$\mathbb{R}^m$$ be given. Show that there always exists a linear transformation $$T$$ such that $$T(A_i) = B_i$$.

## Exercise $$\PageIndex{101}$$

Find the matrix for $$T (\vec{w}) = \text{proj}_{\vec{v}} (\vec{w})$$ where $$\vec{v}=\left[\begin{array}{ccc}1&-2&3\end{array}\right]^T$$.

Recall that $$\text{proj}_{\vec{u}}(\vec{v}) = \frac{\vec{v}\bullet\vec{u}}{||\vec{u}||^2}\vec{u}$$ and so the desired matrix has $$i$$th column equal to $$\text{proj}_{\vec{u}} (\vec{e}_i)$$. Therefore, the matrix desired is $\frac{1}{14}\left[\begin{array}{rrr}1&-2&3\\-2&4&-6\\3&-6&9\end{array}\right]\nonumber$

## Exercise $$\PageIndex{102}$$

Find the matrix for $$T (\vec{w}) = \text{proj}_{\vec{v}} (\vec{w})$$ where $$\vec{v}=\left[\begin{array}{ccc}1&5&3\end{array}\right]^T$$.

$\frac{1}{35}\left[\begin{array}{ccc}1&5&3\\5&25&15\\3&15&9\end{array}\right]\nonumber$

## Exercise $$\PageIndex{103}$$

Find the matrix for $$T (\vec{w}) = \text{proj}_{\vec{v}} (\vec{w})$$ where $$\vec{v}=\left[\begin{array}{ccc}1&0&3\end{array}\right]^T$$.

$\frac{1}{10}\left[\begin{array}{ccc}1&0&3\\0&0&0\\3&0&9\end{array}\right]\nonumber$

## Exercise $$\PageIndex{104}$$

Let $$B=\left\{\left[\begin{array}{r}2\\-1\end{array}\right],\:\left[\begin{array}{c}3\\2\end{array}\right]\right\}$$ be a basis of $$\mathbb{R}^2$$ and let $$\vec{x}=\left[\begin{array}{r}5\\-7\end{array}\right]$$ be a vector in $$\mathbb{R}^2$$. Find $$C_B(\vec{x})$$.

## Exercise $$\PageIndex{105}$$

Let $$B=\left\{\left[\begin{array}{r}1\\-1\\2\end{array}\right],\:\left[\begin{array}{c}2\\1\\2\end{array}\right],\:\left[\begin{array}{r}-1\\0\\2\end{array}\right]\right\}$$ be a basis of $$\mathbb{R}^3$$ and let $$\vec{x}=\left[\begin{array}{r}5\\-1\\4\end{array}\right]$$ be a vector in $$\mathbb{R}^2$$. Find $$C_B(\vec{x})$$.

$$C_B(\vec{x})=\left[\begin{array}{r}2\\1\\-1\end{array}\right]$$.

## Exercise $$\PageIndex{106}$$

Let $$T$$: $$\mathbb{R}^2\mapsto \mathbb{R}^2$$ be a linear transformation defined by $$T\left(\left[\begin{array}{c}a\\b\end{array}\right]\right)=\left[\begin{array}{c}a+b\\a-b\end{array}\right]$$.

Consider the two bases $B_1=\{\vec{v}_1,\vec{v}_2\}=\left\{\left[\begin{array}{c}1\\0\end{array}\right],\:\left[\begin{array}{r}-1\\1\end{array}\right]\right\}\nonumber$ and $B_2=\left\{\left[\begin{array}{c}1\\1\end{array}\right],\:\left[\begin{array}{r}1\\-1\end{array}\right]\right\}\nonumber$ Find the matrix $$M_{B_2,B_1}$$ of $$T$$ with respect to the bases $$B_1$$ and $$B_2$$.

$$M_{B_2B_1}=\left[\begin{array}{rr}1&0\\-1&1\end{array}\right]$$