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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