7.E: Exercises for Chapter 7

Calculational Exercises

1. Let \(T \in \cal{L}(\mathbb{F}^2 , \mathbb{F}^2)\) be defined by

\[T (u, v) = (v, u)\]

for every \(u, v \in \mathbb{F}.\) Compute the eigenvalues and associated eigenvectors for \(T.\)

2. Let \(T \in \cal{L}(\mathbb{F}^3 , \mathbb{F}^3\)) be defined by

\[T (u, v, w) = (2v, 0, 5w)\]

for every \(u, v, w \in \mathbb{F}.\) Compute the eigenvalues and associated eigenvectors for \(T.\)

3. Let \(n \in \mathbb{Z}_+\) be a positive integer and \(T \in \cal{L}(\mathbb{F}^n , \mathbb{F}^n )\) be defined by

\[T (x_1 ,\ldots, x_n ) = (x1 + \cdots + x_n , \ldots, x_1 + \cdots + x_n)\]

for every \(x_1 ,\ldots, x_n \in \mathbb{F}.\) Compute the eigenvalues and associated eigenvectors for \(T.\)

4. Find eigenvalues and associated eigenvectors for the linear operators on \(\mathbb{F}^2\) defined by each given \(2 \times 2\) matrix.

\((a) \left[ \begin{array}{cc} 3 & 0 \\ 8 & -1 \end{array} \right], ~~ (b) \left[ \begin{array}{cc} 10 & -9 \\ 4 & -2 \end{array} \right], ~~ (c) \left[ \begin{array}{cc} 0 & 3 \\ 4 & 0 \end{array} \right], \)

\((d) \left[ \begin{array}{cc} -2 & -7 \\ 1 & 2 \end{array} \right], ~~ (e) \left[ \begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array} \right], ~~ (f) \left[ \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right] \)


Hint: Use the fact that, given a matrix \(A = \left[ \begin{array}{cc} a & b \\ c & d \end{array} \right] \in \mathbb{F}^{2\times2} , \lambda \in \mathbb{F}\) is an eigenvalue for A if and only if \((a − \lambda)(d − \lambda) − bc = 0.\)

5. For each matrix \(A\) below, find eigenvalues for the induced linear operator \(T\) on \(\mathbb{F}^n\) without performing any calculations. Then describe the eigenvectors \(v \in \mathbb{F}^n\) associated to each eigenvalue \(\lambda\) by looking at solutions to the matrix equation \((A − \lambda I)v = 0,\) where I denotes the identity map on \(\mathbb{F}^n.\)

\((a) \left[ \begin{array}{cc} -1 & 6 \\ 0 & 5 \end{array} \right], ~~(b) \left[ \begin{array}{cccc} -\frac{1}{3} & 0 & 0 & 0 \\ 0 & -\frac{1}{3} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & \frac{1}{2} \end{array} \right], ~~(c) \left[ \begin{array}{cccc} 1 & 3 & 7 & 11 \\ 0 & \frac{1}{2} & 3 & 8 \\ 0 & 0 & 0 & 4 \\ 0 & 0 & 0 & 2 \end{array} \right]\)

6. For each matrix \(A\) below, describe the invariant subspaces for the induced linear operator \(T on \mathbb{F}^2\) that maps each \(v \in \mathbb{F}^2\) to \(T (v) = Av.\)

\((a) \left[ \begin{array}{cc} 4 & -1 \\ 2 & 1 \end{array} \right], ~~ (b) \left[ \begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array} \right], ~~ (c) \left[ \begin{array}{cc} 2 & 3 \\ 0 & 2 \end{array} \right], ~~ (d) \left[ \begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array} \right] \)

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

\[ T \left( \begin{array}{c} x \\ y \end{array} \right) = \left( \begin{array}{c} x \\ x+y \end{array} \right),~~\rm{~for ~all~} \left( \begin{array}{c} x \\ y \end{array} \right) \in \mathbb{R}^2.\]

Define two real numbers \(\lambda_+\) and \(\lambda_−\) as follows:

\[ \lambda_+=\frac{1+\sqrt{5}}{2}, ~~ \lambda_-=\frac{1-\sqrt{5}}{2}.\]

(a) Find the matrix of \(T\) with respect to the canonical basis for \(\mathbb{R}^2\) (both as the domain and the codomain of \(T\) ; call this matrix \(A\)).
(b) Verify that \(\lambda_+\) and \(\lambda_−\) are eigenvalues of \(T\) by showing that \(v_+\) and \(v_−\) are eigen-
vectors, where

\[ v_+ = \left( \begin{array}{c} 1 \\ \lambda_+ \end{array} \right),~~ v_- = \left( \begin{array}{c} 1 \\ \lambda_- \end{array} \right).\]

(c) Show that \((v_+ , v_− )\) is a basis of \(\mathbb{R}^2.\)
(d) Find the matrix of \(T\) with respect to the basis \((v_+ , v_− )\) for \(\mathbb{R}^2\) (both as the domain
and the codomain of \(T\) ; call this matrix \(B\)).

Proof-Writing Exercises

1. Let \(V\) be a finite-dimensional vector space over \(\mathbb{F}\) with \(T \in \cal{L}\)\((V, V),\) and let \(U_1 , \ldots, U_m\) be subspaces of \(V\) that are invariant under \(T \). Prove that \(U_1 + \cdots + U_m\) must then also be an invariant subspace of \(V\) under \(T .\)

2. Let \(V\) be a finite-dimensional vector space over \(\mathbb{F }\) with \(T \in \cal{L}\)\((V, V ),\) and suppose that \(U_1\) and \(U_2\) are subspaces of \(V\) that are invariant under \(T \). Prove that \(U_1 \cap U_2\) is also an invariant subspace of \(V\) under \(T.\)

3. Let \(V\) be a finite-dimensional vector space over \(\mathbb{F}\) with \(T \in \cal{L}\)\((V, V)\) invertible and \(\lambda \in \mathbb{F} = \{0\}.\) Prove \(\lambda\) is an eigenvalue for \(T\) if and only if \(\lambda−1\) is an eigenvalue for \(T −1.\)

4. Let \(V\) be a finite-dimensional vector space over \(\mathbb{F},\) and suppose that \(T \in \cal{L}\)\((V, V)\) has the property that every \(v \in V\) is an eigenvector for \(T\) . Prove that \(T\) must then be a scalar multiple of the identity function on \(V.\)

5. Let \(V\) be a finite-dimensional vector space over \(\mathbb{F},\) and let \(S, T \in \cal{L}\)\((V)\) be linear operators on \(V\) with \(S\) invertible. Given any polynomial \(p(z) \in \mathbb{F}[z],\) prove that

\[p(S \circ T \circ S^{ −1} ) = S \circ p(T ) \circ S^{ −1}.\]

6. Let \(V\) be a finite-dimensional vector space over \(\mathbb{C}, T \in \cal{L}\)\((V)\) be a linear operator on \(V\) , and \(p(z) \in \mathbb{C}[z]\) be a polynomial. Prove that \(\lambda \in \mathbb{C}\) is an eigenvalue of the linear operator \(p(T ) \in \cal{L}\)\((V)\) if and only if \(T\) has an eigenvalue \(\mu \in \mathbb{C}\) such that \(p(\mu) = \lambda.\)

7. Let \(V\) be a finite-dimensional vector space over \(\mathbb{C}\) with \(T \in \cal{L}\)\((V)\) a linear operator on \(V.\) Prove that, for each \(k = 1,\ldots , dim(V ),\) there is an invariant subspace \(U_k\) of \(V\) under \(T\) such that \(dim(U_k ) = k.\)

8. Prove or give a counterexample to the following claim:

Claim. Let \(V\) be a finite-dimensional vector space over \(\mathbb{F},\) and let \(T \in \cal{L}\)\((V)\) be a linear operator on \(V\) . If the matrix for \(T\) with respect to some basis on \(V\) has all zeros on the diagonal, then \(T\) is not invertible.

9. Prove or give a counterexample to the following claim:

Claim. Let \(V\) be a finite-dimensional vector space over \(\mathbb{F},\) and let \(T \in \cal{L}\)\((V)\) be a linear
operator on \(V\) . If the matrix for \(T\) with respect to some basis on \(V\) has all non-zero elements on the diagonal, then \(T\) is invertible.

10. Let \(V\) be a finite-dimensional vector space over \(\mathbb{F},\) and let \(S, T \in \cal{L}\)\((V)\) be linear operators on \(V\) . Suppose that \(T\) has \(dim(V)\) distinct eigenvalues and that, given any eigenvector \(v \in V\) for \(T\) associated to some eigenvalue \(\lambda \in \mathbb{F},\) \(v\) is also an eigenvector
for \(S\) associated to some (possibly distinct) eigenvalue \(\mu \in \mathbb{F}.\) Prove that \(T \circ S = S \circ T .\)

11. Let \(V\) be a finite-dimensional vector space over \(\mathbb{F},\) and suppose that the linear operator \(P \in \cal{L}\)\((V)\) has the property that \(P^2 = P.\) Prove that \(V = null(P ) \oplus range(P ).\)

12. (a) Let \(a, b, c, d \in \mathbb{F}\) and consider the system of equations given by

\[ax_1 + bx_2 = 0 \\ cx_1 + dx_2 = 0.\]

Note that \(x_1 = x_2 = 0\) is a solution for any choice of \(a, b, c,\) and \(d\). Prove that this system of equations has a non-trivial solution if and only if \(ad − bc = 0.\)

(b) Let \(A = \left[ \begin{array}{cc} a & b \\ c & d \end{array} \right] \in \mathbb{F}^{2\times2}\) , and recall that we can define a linear operator \(T \in \cal{L}(\mathbb{F}^2 )\) on \(\mathbb{F}^2\) by setting \(T (v) = Av\) for each \(v = \left[ \begin{array}{c} v_1 \\ v_2 \end{array} \right] \in \mathbb{F}^2. \)

Show that the eigenvalues for \(T\) are exactly the \(\lambda \in \mathbb{F}\) for which \(p(\lambda) = 0,\) where \(p(z) = (a − z)(d − z) − bc.\)

Hint: Write the eigenvalue equation \(Av = \lambda v\) as \((A − \lambda I)v = 0\) and use the first part.

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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