Find a basis of $U_i$, given subspace $U_1$,$U_2$?

Question

I've been given two subspaces, $U_1$, $U_2$, which are as follows :

$ U_1=\left\{\begin{pmatrix} a&b \\c&d\\ \end{pmatrix}: b=0\right\}$

$ U_2=\left\{\begin{pmatrix} a&b \\c&d\\ \end{pmatrix}: a=c=d\right\}$

I need to find a basis for $U_i$?

Thanks for your help!

Answer 1

For $U_1$: $$ \begin{pmatrix} a & 0 \\ c & d \end{pmatrix} = a\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} + c\begin{pmatrix} ? & ? \\ ? & ? \end{pmatrix} + d\begin{pmatrix} ? & ? \\ ? & ? \end{pmatrix} $$

For $U_2$: $$ \begin{pmatrix} a & b \\ a & a \end{pmatrix} = a\begin{pmatrix} ? & ? \\ ? & ? \end{pmatrix} + b\begin{pmatrix} ? & ? \\ ? & ? \end{pmatrix} $$

Answer 2

Hint 1

The number of free parameters will be the dimension, that is, the number of elements in a basis.

Hint 2

Write a generic matrix of $U_{i}$ as a sum of matrices, such that a given parameter occurs in one and only one of the summands.

Hint 3

Now take the parameter out in each summand, meaning writing for instance $$\begin{bmatrix}a & 0\\ 0 & 0\end{bmatrix} = a \begin{bmatrix}1 & 0\\ 0 & 0\end{bmatrix}$$