Exercise 23

(Schur’s Triangularization Lemma)

• Let $U$ be a subspace of $V$ and fix bases $B_{U}\subseteq B_{V}$. What is the relationship between the representation of a vector from $U$ with respect to $B_{U}$ and the representation of that vector (viewed as a member of $V$) with respect to $B_{V}$?

• What about maps?

• Fix a basis $B=\langle\vec{\beta}_{1},\dots,\vec{\beta}_{n}\rangle$ for $V$ and observe that the spans

  $$[\varnothing]=\text{$\{\hskip 1.00006pt\vec{0}\hskip 1.00006pt\}$}\subset[\text{$\{\hskip 1.00006pt\vec{\beta}_{1}\hskip 1.00006pt\}$}]\subset[\text{$\{\hskip 1.00006pt\vec{\beta}_{1},\vec{\beta}_{2}\hskip 1.00006pt\}$}]\subset\quad\cdots\quad\subset[B]=V$$

  form a strictly increasing chain of subspaces. Show that for any linear map $h\colon V\to W$ there is a chain $W_{0}=\text{$\{\hskip 1.00006pt\vec{0}\hskip 1.00006pt\}$}\subseteq W_{1}\subseteq\dots\subseteq W_{m}=W$ of subspaces of $W$ such that

  $$h([\text{$\{\hskip 1.00006pt\vec{\beta}_{1},\dots,\vec{\beta}_{i}\hskip 1.00006pt\}$}])\subseteq W_{i}$$

  for each $i$.

• Conclude that for every linear map $h\colon V\to W$ there are bases $B,D$ so the matrix representing $h$ with respect to $B,D$ is upper-triangular (that is, each entry $h_{i,j}$ with $i>j$ is zero).

• Is an upper-triangular representation unique?