Let L be a linear operator on a vector space V of dimension 5 and let A be any matrix representing L. If L is nilpotent of index 3, then what are the possible Jordan canonical forms of A?

Let A be a $4\times 4$ matrix whose only eigenvalue is $\lambda =2$. What are the possible Jordan canonical forms of A?

Let L be a linear operator on a vector space V of dimension 6 and let A be a matrix representing L. If L has only one distinct eigenvalue $\lambda$ and the eigenspace $S_{\lambda }$ has dimension 3, then what are the possible Jordan canonical forms of A?

For each of the following, find a matrix S such that $S^{-1}\mathit{\text{AS}}$ is a simple Jordan matrix:

1. $A=\left[\begin{array}{ccc}1& 0& 1\\ 1& 0& 2\\ 1& -1& 2\end{array}\right]$

2. $A=\left[\begin{array}{cccc}1& 2& 0& 0\\ 0& 1& 2& 0\\ 0& 0& 1& 2\\ 0& 0& 0& 1\end{array}\right]$

For each of the following, find a matrix S such that $S^{-1}\mathit{\text{AS}}$ is the Jordan canonical form of A:

1. $A=\left[\begin{array}{cccc}-1& 1& 0& 0\\ -1& 1& 0& 0\\ -2& 2& 0& 0\\ 0& 3& -1& 0\end{array}\right]$

2. $A=\left[\begin{array}{ccccc}0& 0& 1& 1& 1\\ 0& 0& 0& 1& 1\\ 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\end{array}\right]$

Let $S_1$ and $S_2$ be subspaces of a finite dimensional vector space V. Prove that $V=S_1\oplus S_2$ if and only if $V=S_1+S_2$ and $S_1\cap S_2=\left\{0\right\}$.

Prove Lemma 8.1.1.

Let L be a linear operator mapping a vector space V into itself. Show that ker(L)and R(L) are invariant subspaces of V under L.

Let L be a linear operator on a vector space V. Let $S_k\left[\text{v}\right]$ denote the subspace spanned by $\text{v},L\left(\text{v}\right),\dots ,L^{k-1}\left(\text{v}\right)$. Show that $S_k\left[\mathbf{v}\right]$ is invariant under L if and only if $L^k\left(\mathbf{v}\right)\in S_k\left[\mathbf{v}\right]$.

Let L be a linear operator on a vector space V and let S be a subspace of V. Let $⌶$ represent the identity operator and let $\lambda$ be a scalar. Show that L is invariant on S if and only if $L-\lambda ⌶$ is invariant on S.

Let S be the subspace of $C[a,b]$ spanned by $x,{\mathit{\text{xe}}}^x$, and ${\mathit{\text{xe}}}^x+x^2{e}^x$. Let D be the differentiation operator on S.

1. Find a matrix A representing D with respect to $[e^x,{\mathit{\text{xe}}}^x,{\mathit{\text{xe}}}^x+x^2{e}^x]$.

2. Determine the Jordan canonical form of A and the corresponding basis of S.

Let D denote the linear operator on $P_n$ defined by $D\left(p\right)=p\prime$ for all $p\in P_n$. Show that D is nilpotent and can be represented by a simple Jordan matrix.