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# 6.E: Exercises for Chapter 6

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## Calculational Exercises

1. Define the map $$T: \mathbb{R}^2 \to \mathbb{R}^2$$ by $$T(x,y)=(x+y,x)$$.

1. Show that $$T$$ is linear.
2. Show that $$T$$ is surjective.
3. Find $$\dim\left(\text{null}\left(T\right)\right)$$.
4. Find the matrix for $$T$$ with respect to the canonical basis of $$\mathbb{R}^2$$.
5. Find the matrix for $$T$$ with respect to the canonical basis for the domain $$\mathbb{R}^2$$ and the basis $$((1,1),(1,-1))$$ for the target space $$\mathbb{R}^2$$.
6. Show that the map $$F:\mathbb{R}^2 \to \mathbb{R}^2$$ given by $$F(x,y)=(x+y,x+1)$$ is not linear.

2. Let $$T\in\mathcal{L}(\mathbb{R}^2)$$ be defined by

$T\begin{pmatrix} x\\ y\end{pmatrix} = \begin{pmatrix}y\\ -x\end{pmatrix},\quad \mbox{ for all } \begin{pmatrix}x\\ y\end{pmatrix}\in \mathbb{R}^2.$

1. Show that $$T$$ is surjective.
2. Find $$\dim\left(\text{null}\left(T\right)\right)$$.
3. Find the matrix for $$T$$ with respect to the canonical basis of $$\mathbb{R}^2$$.
4. Show that the map $$F:\mathbb{R}^2 \to \mathbb{R}^2$$ given by $$F(x,y)=(x+y,x+1)$$ is not linear.

3. Consider the complex vector spaces $$\mathbb{C}^2$$ and $$\mathbb{C}^3$$ with their canonical bases, and define $$S \in \mathcal{L}(\mathbb{C}^3,\mathbb{C}^2)$$ be the linear map defined by $$S(v) = A v, \forall v \in \mathbb{C}^{3}$$, where $$A$$ is the matrix

$A = M(S) = \begin{pmatrix} i &1 &1 \\ 2i& -1& -1 \end{pmatrix} .$

Find a basis for $$null(S).$$

4. Give an example of a function $$f: \mathbb{R}^{2} \to \mathbb{R}$$ having

the property that

$\forall a \in \mathbb{R}, \forall v \in \mathbb{R}^2, f(av) = a f(v)$

but such that $$f$$ is not a linear map.

5. Show that the linear map $$T: \mathbb{F}^{4} \to \mathbb{F}^{2}$$ is surjective if

$\mbox{null}(T) = \{(x_{1}, x_{2}, x_{3}, x_{4}) \in \mathbb{F}^{4} \ | \ x_{1} = 5 x_{2}, x_{3} = 7 x_{4} \}.$

6. Show that no linear map $$T: \mathbb{F}^{5} \to \mathbb{F}^{2}$$ can

have as its null space the set

$\{(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}) \in \mathbb{F}^{5} \ | \ x_{1} = 3 x_{2}, x_{3} = x_{4} = x_{5} \}.$

7. Describe the set of solutions $$x=(x_1,x_2,x_3)\in\mathbb{R}^3$$ of the system of equations

$\left. \begin{array}{rl} x_1-x_2+x_3&=0 \\ x_1+2x_2 +x_3&=0 \\ 2x_1+x_2+2x_3&=0 \end{array} \right\}.$

## Proof-Writing Exercises

1. Let $$V$$ and $$W$$ be vector spaces over $$\mathbb{F}$$ with $$V$$ finite-dimensional, and let $$U$$ be any
subspace of $$V$$ . Given a linear map $$S \in \cal{L}(U,W),$$ prove that there exists a linear map
$$T \in \cal{L}(V,W)$$ such that, for every $$u \in U, S(u) = T(u).$$

2. Let $$V$$ and $$W$$ be vector spaces over $$\mathbb{F},$$ and suppose that $$T \in \cal{L}(V,W)$$ is injective.
Given a linearly independent list $$(v_1,\ldots , v_n)$$ of vectors in $$V$$, prove that the
list $$(T(v_1), \ldots ,T(v_n))$$ is linearly independent in $$W.$$

3. Let $$U, V,$$ and $$W$$ be vector spaces over $$\mathbb{F},$$ and suppose that the linear maps $$S \in \cal{L}(U, V )$$
and $$T \in \cal{L}(V,W)$$ are both injective. Prove that the composition map $$T \circ S$$ is injective.

4. Let $$V$$ and $$W$$ be vector spaces over $$\mathbb{F},$$ and suppose that $$T \in \cal{L}(V,W)$$ is surjective.
Given a spanning list $$(v_1,\ldots , v_n)$$ for $$V$$ , prove that

$span(T(v_1),\ldots ,T(v_n)) = W.$

5. Let $$V$$ and $$W$$ be vector spaces over $$\mathbb{F}$$ with $$V$$ finite-dimensional. Given $$T \in \cal{L}(V,W),$$
prove that there is a subspace $$U$$ of $$V$$ such that

$U \cap null(T) = \{0\} \rm{~and~} range(T) = \{T(u) | u \in U\}.$

6. Let $$V$$ be a vector space over $$\mathbb{F},$$ and suppose that there is a linear map $$T \in \cal{L}(V, V )$$
such that both $$null(T)$$ and $$range(T)$$ are finite-dimensional subspaces of $$V$$ . Prove that
$$V$$ must also be finite-dimensional.

7. Let $$U, V,$$ and $$W$$ be finite-dimensional vector spaces over $$\mathbb{F}$$ with $$S \in \cal{L}(U, V )$$ and
$$T \in \cal{L}(V,W).$$ Prove that

$dim(null(T \circ S)) \leq dim(null(T)) + dim(null(S)).$

8. Let $$V$$ be a finite-dimensional vector space over $$\mathbb{F}$$ with $$S, T \in \cal{L}(V, V).$$ Prove that
$$T \circ S$$ is invertible if and only if both $$S$$ and $$T$$ are invertible.

9. Let $$V$$ be a finite-dimensional vector space over $$\mathbb{F}$$ with $$S, T \in \cal{L}(V, V ),$$ and denote by
I the identity map on $$V$$ . Prove that $$T \circ S = I$$ if and only if $$S \circ T = I.$$

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