Check if the set is a subspace of vector space, determine a basis and its dimension

Question

1. Check if the given set $U\subseteq V$ is a subspace of space $V$.

2. If it is, determine its basis and dimension.

3. If you say that some set is a basis, you must prove it.

a) $U=\{(2a,4b,a-3b,4a+5b:a,b\in\mathbb R)\},V=\mathbb R^4$

b) $U=\{(A\in V:S^{-1}AS \text{ is diagonal matrix})\},V= M_n$

Could you please show the steps for solving this problem?

Answer 1

Are all $u \in U$ elements of $V$

If $u,v$ are in $U$ is $u+v$ and $\alpha u$ in $U$?

Is $0$ in $U$?

a) For a basis there is one basis associated with $a$ and one associated with $b$

Show that these are linearly independent and span $U$

b) Is a little trickier but you should with some effort you should be able to find, $u,v$ such that $u+v$ is not diagonalizable.

Actually, I may be reading b) incorrectly. Is it saying that $S$ is a given matrix in $M_n$ or there exists an $S \in M_n: S^{-1} A S$ is diagonal?

Answer 2

I'll do b). Leave a) for you.

$U$ isn't a subspace, for any n... The sum of two diagonalizable $A$ and $B$ is not necessarily diagonalizable. E.g. let $A=\begin{pmatrix}1&0\\0&0\end{pmatrix}$ and $B=\begin{pmatrix} 0&1\\0&1\end{pmatrix}$. This is a counterexample for $n=2$. There are also counterexamples for $n\gt2$.