Linear algebra
1. Let $k$ be a field, char $k \neq 2$. For $n \geq 2$, show that there is a basis of $M_{n}(k)$ (the ring of $n \times n$ matrices over $k$ ) consisting only of non-diagonalizable matrices.
2. Let $k$ be a field, and $D: M_{n}(k) \rightarrow k$ a multiplicative function, i.e. $D(A B)=D(A) D(B)$ for all $A, B \in M_{n}(k)$.
i) Show that the following are equivalent:
a) $D(0) \neq D(I)$
b) $D(0)=0, D(I)=1$
c) $D$ vanishes on a proper nonempty subset of $M_{n}(k)$.
ii) If the conditions in (i) are satisfied, show that the vanishing set of $D$ is precisely the set of singular matrices, i.e. matrices with determinant 0 . Deduce that the set of maps $D$ satisfying (i) is in bijection with group homomorphisms from $G L_{n}(k)$ to $k^{\times}$.
3. i) Let $\varphi$ be an endomorphism of a finite-dimensional $k$-vector space. Suppose $g, h \in k[x]$ satisfy $(g, h)=1, g(\varphi) h(\varphi)=0$. Show that $\ker g(\varphi) \cap \operatorname{im} g(\varphi)=0$.
ii) Show that one can always satisfy the hypotheses of (i) by taking $g=x^{n}$ for some $h \in k[x]$, $n \geq \dim \ker \varphi$.
4. Let $V$ be a vector space, $f_{i} \in \operatorname{End}(V)$ pairwise commuting endomorphisms, and $E_{i}:=\ker f_{i}$. Show that $\sum_{i} E_{i}=\bigoplus_{i} E_{i}$ iff $E_{i} \cap E_{j}=0$ for all $i \neq j$.
5. Let $V$ a finite-dimensional $\mathbb{Q}$-vector space. Show that there $\operatorname{exists} \varphi \in \operatorname{End}(V)$ with no nontrivial proper invariant subspaces.
Past exam problems
6. (6.10.2) Let $k$ be a field. Show that the only two-sided ideals of $M_{n}(k)$ are 0 and $M_{n}(k)$.
7. (7.7.7) Let $A \in G L_{2}(\mathbb{Z})$. If $A^{n}=I$ for some $n$, show that $A^{12}=I$.
8. (7.7.12) Let $A \in M_{n}(\mathbb{C})$. Is $A$ similar to $A^{t}$ ?
9. (7.4.31) Let $V$ be the vector space of polynomials in one variable over $\mathbb{R}$ of degree $\leq 10$, and let $D: V \rightarrow V$ be differentiation.
i) Show that $\operatorname{tr} D=0$.
ii) Find all eigenvectors of $D$ and $e^{D}$.
10. (7.5.21) If $A \in M_{n}(\mathbb{C})$, and $f \in \mathbb{C}[x]$, show that the eigenvalues of $f(A)$ are precisely $f(\lambda)$, where $\lambda$ is an eigenvalue of $A$.
1. Let $k$ be a field, char $k \neq 2$. For $n \geq 2$, show that there is a basis of $M_{n}(k)$ (the ring of $n \times n$ matrices over $k$ ) consisting only of non-diagonalizable matrices.
2. Let $k$ be a field, and $D: M_{n}(k) \rightarrow k$ a multiplicative function, i.e. $D(A B)=D(A) D(B)$ for all $A, B \in M_{n}(k)$.
i) Show that the following are equivalent:
a) $D(0) \neq D(I)$
b) $D(0)=0, D(I)=1$
c) $D$ vanishes on a proper nonempty subset of $M_{n}(k)$.
ii) If the conditions in (i) are satisfied, show that the vanishing set of $D$ is precisely the set of singular matrices, i.e. matrices with determinant 0 . Deduce that the set of maps $D$ satisfying (i) is in bijection with group homomorphisms from $G L_{n}(k)$ to $k^{\times}$.
3. i) Let $\varphi$ be an endomorphism of a finite-dimensional $k$-vector space. Suppose $g, h \in k[x]$ satisfy $(g, h)=1, g(\varphi) h(\varphi)=0$. Show that $\ker g(\varphi) \cap \operatorname{im} g(\varphi)=0$.
ii) Show that one can always satisfy the hypotheses of (i) by taking $g=x^{n}$ for some $h \in k[x]$, $n \geq \dim \ker \varphi$.
4. Let $V$ be a vector space, $f_{i} \in \operatorname{End}(V)$ pairwise commuting endomorphisms, and $E_{i}:=\ker f_{i}$. Show that $\sum_{i} E_{i}=\bigoplus_{i} E_{i}$ iff $E_{i} \cap E_{j}=0$ for all $i \neq j$.
5. Let $V$ a finite-dimensional $\mathbb{Q}$-vector space. Show that there $\operatorname{exists} \varphi \in \operatorname{End}(V)$ with no nontrivial proper invariant subspaces.
Past exam problems
6. (6.10.2) Let $k$ be a field. Show that the only two-sided ideals of $M_{n}(k)$ are 0 and $M_{n}(k)$.
7. (7.7.7) Let $A \in G L_{2}(\mathbb{Z})$. If $A^{n}=I$ for some $n$, show that $A^{12}=I$.
8. (7.7.12) Let $A \in M_{n}(\mathbb{C})$. Is $A$ similar to $A^{t}$ ?
9. (7.4.31) Let $V$ be the vector space of polynomials in one variable over $\mathbb{R}$ of degree $\leq 10$, and let $D: V \rightarrow V$ be differentiation.
i) Show that $\operatorname{tr} D=0$.
ii) Find all eigenvectors of $D$ and $e^{D}$.
10. (7.5.21) If $A \in M_{n}(\mathbb{C})$, and $f \in \mathbb{C}[x]$, show that the eigenvalues of $f(A)$ are precisely $f(\lambda)$, where $\lambda$ is an eigenvalue of $A$.