9.E: Exercises

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.

Answer

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 $\mathbb{R}^n$.

Answer

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.

Answer

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 $span\{\vec{x}_1,\cdots ,\vec{x}_k\}$.

Answer

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

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 $\mathbb{R}^5$?

Answer

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.

Answer

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. $\{ x^3 +1,\: x^2 +x,\: 2x^3 +x^2 ,\: 2x^3 −x^2 −3x+1\}$

Answer

2. 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.

Answer

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.

Answer

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.

Answer

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 $\mathbb{R}^4$?

Answer

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 $\mathbb{R}^4$?

Answer

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 $T(ax^2+bx+c)$.

Answer

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 $T(\vec{x})=A\vec{x}$.

Answer

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

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 $T(\vec{x})=A\vec{x}$.

Answer

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

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.

Answer

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.

Answer

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 $S$.

Answer

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 $Im(T)$.

Answer

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 $Im(T)$.

Answer

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 $V$.

Answer

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 $\text{dim}(\text{ker}(T))$?

Answer

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 $T (\vec{x}) = A\vec{x}$.

Answer

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 $$\left[\begin{array}{ccc}B_1&\cdots&B_n\end{array}\right]\:\left[\begin{array}{ccc}A_1&\cdots&A_n\end{array}\right]^{-1}\nonumber$$

Answer

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 $T(\vec{x}) = A\vec{x}$.

Answer

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

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 $T(\vec{x}) = A\vec{x}$.

Answer

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

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 $T(\vec{x}) = A\vec{x}$.

Answer

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

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 $T(\vec{x}) = A\vec{x}$.

Answer

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

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 $T(\vec{x}) = A\vec{x}$.

Answer

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

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 $\vec{v}=\left[\begin{array}{ccc}1&-2&3\end{array}\right]^T$.

Answer

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 $\vec{v}=\left[\begin{array}{ccc}1&5&3\end{array}\right]^T$.

Answer

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

Exercise $\PageIndex{103}$

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

Answer

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

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 $C_B(\vec{x})$.

Answer

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

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 $B_2$.

Answer

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