6.E: Exercises for Chapter 6

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 $S \circ T = I.$

Contributors

• Isaiah Lankham, Mathematics Department at UC Davis
• Bruno Nachtergaele, Mathematics Department at UC Davis
• Anne Schilling, Mathematics Department at UC Davis

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