Linear Algebra Exercises

• The following are exercises in linear algebra.

Exercise 1

• Let $E$ be a vector space over $\mathbb{K}$ and $u \in \mathcal{L}(E)$.
• There exists an $x \in E$ such that the family $\left(x, u(x), \ldots, u^{n-1}(x)\right)$ is free.
• $\left(x, u(x), \ldots, u^{n}(x)\right)$ is linked.
• The family $\left(I d, u, \ldots, u^{n-1}\right)$ is free.

Exercise 2

• Let $f \in \mathcal{L}\left(\mathbb{R}^{3}\right)$ be defined by $f(x, y, z)=(x+y+z, x+y+z, x+y+z)$.
• Determine a basis of $\operatorname{Ker}(f)$ and $\operatorname{Im}(f)$.
• Check whether $\operatorname{Ker}(f) \oplus \operatorname{Im}(f)=\mathbb{R}^{3}$.

Exercise 3

• Let $E$ be a $\mathbb{K}$-vector space of finite dimension and $u \in \mathcal{L}(E)$ such that $\operatorname{rg}(u)=1$.
• Prove either $u^{2}=0$ or $u^{2}=\operatorname{tr}(u) u$.
• In the latter case, $u$ is diagonalizable.

Exercise 4

• Let $E$ be a $\mathbb{K}$-vector space of finite dimension.
• Let $u, v \in \mathcal{L}(E)$ such that $u \circ v=0$.
• Show that $\operatorname{rg}(u)+\operatorname{rg}(v) \leq \operatorname{dim}(E)$.

Exercise 5

• Let $E$ be a $\mathbb{K}$-vector space of finite dimension.
• Let $u, v \in \mathcal{L}(E)$ such that $u+v=I d_{E}$ and $u \circ v=v \circ u=0$.
• Show that $u$ and $v$ are projectors.
• Determine their images and their kernels.

Exercise 6

• Let $p$ and $q$ be two projectors of a vector space $E$.
• Show that $p+q$ is a projector if and only if $p \circ q=q \circ p=0$.

Exercise 7

• Let $E$ be a vector space of finite dimension.
• Show that any rank 1 operator $u \in \mathcal{L}(E)$ is similar to a matrix with all entries zero, except for the first entry of the first row.

Exercise 8

• Let $u \in \mathcal{L}\left(\mathbb{R}^{3}\right)$ be defined by $u(x, y, z)=(z, x, y)$.
• Determine the eigenvalues of $u$.
• Check whether $u$ is diagonalizable.

Exercise 9

• Let $A \in \mathcal{M}_{n}(\mathbb{R})$ be the matrix whose entries are all 1.
• Determine the eigenvalues of $A$.
• Check whether $A$ is diagonalizable.

Exercise 10

• Let $A=\left(\begin{array}{ccc}5 & -3 & 2 \ 6 & -4 & 4 \ 4 & -4 & 5\end{array}\right)$.
• Determine the eigenvalues of $A$.
• Determine the eigenspaces of $A$.
• Check whether the matrix $A$ is diagonalizable.
• If so, provide an invertible matrix $P$ and a diagonal matrix $D$ such that $A=P D P^{-1}$.

Exercise 11

• Let $A=\left(\begin{array}{ccc}2 & 1 & 1 \ 1 & 2 & 1 \ 1 & 1 & 2\end{array}\right)$.
• Determine an orthogonal matrix $P$ such that ${ }^{t} P A P$ is diagonal.

Exercise 12

• Let $A, B \in \mathcal{M}_{n}(\mathbb{R})$.
• Show that $A B$ and $B A$ have the same eigenvalues.

Exercise 13

• Let $A \in \mathcal{M}_{n}(\mathbb{R})$ be a nilpotent matrix.
• Show that $A$ is similar to a matrix whose diagonal entries are all zero.

Exercise 14

• Let $A \in \mathcal{M}_{n}(\mathbb{C})$.
• Show that $A$ is diagonalizable if and only if $\operatorname{rg}(A)=\operatorname{tr}(A)$.

Exercise 15

• Solve the following differential system:
• $\left{\begin{array}{l}x^{\prime}(t)=5 x(t)-3 y(t)+2 z(t) \ y^{\prime}(t)=6 x(t)-4 y(t)+4 z(t) \ z^{\prime}(t)=4 x(t)-4 y(t)+5 z(t)\end{array}\right.$