# Homework Help: Basis for a Matrix

1. Jun 1, 2013

### Offlinedoctor

I'm having trouble finding the spanning set and basis for the matrix;

| a b |
| c d | with condition that b=d

I'm thinking thinking the spanning set would be
A= x
B = y
C = z

Such that x,y,z are all reals, but I can't think of how to find a basis for this, I'm thinking of doing row echolon form but am thinking of how to set parameters.

2. Jun 1, 2013

### tiny-tim

Hi Offlinedoctor!
I don't understand this at all.

The members of the spanning set will all be matrices.

Try again.

3. Jun 2, 2013

### Skins

Are you looking for a basis for the subspace of all 2x2 matrices such that both entries in the second column are equal ?

Or are you only dealing with a single particular matrix in which case saying "basis for the matrix" would make no sense. Generally when we refer to a basis with regards to a single matrix we are referring to a basis for its column space, row space, or null spaces, of the columns and rows. In the context of linear algebra a basis is a minimal spanning set for a vector space.

Add some more detail to statement of the problem.

4. Jun 2, 2013

### HallsofIvy

I think I answered this before- on a different forum (and for a different poster user name).

I suspect you are asking for a subspace of all 2 by 2 matrices of the form
$$\begin{bmatrix} a & b \\ c & d \end{bmatrix}$$
such that b= d.

Such a matrix looks like
$$\begin{bmatrix}a & b \\ c & b \end{bmatrix}= \begin{bmatrix}a & 0 \\ 0 & 0 \end{bmatrix}+ \begin{bmatrix}0 & b \\ 0 & b \end{bmatrix}+ \begin{bmatrix}0 & 0 \\ c & 0\end{bmatrix}$$
$$= a\begin{bmatrix}1 & 0 \\ 0 & 0 \end{bmatrix}+ b\begin{bmatrix}0 & 1 \\ 0 & 1\end{bmatrix}+ c\begin{bmatrix}0 & 0 \\ 1 & 0 \end{bmatrix}$$