# Dimension of a subspace

1. Mar 28, 2012

### baher

Find the dimension of the subspace of M2;2 consisting of all 2 by 2 matrices
whose diagonal entries are zero. ?

i know that the dimension is the number of vectors that are the basis for this subspace ,but i cannot figure out what is the basis for this subspace ?

any help will be appreciated ,

2. Mar 28, 2012

### Bacle2

Try doing a generalized row-reduction using Gaussian elimination, to see how many parameters determine the subspace.

3. Mar 28, 2012

### homeomorphic

Question:

What's the difference between a 2 by 2 matrix and a column vector with 4 entries?

Presumably, you know how to find a basis for all the column vectors with 4 entries.

4. Mar 28, 2012

### baher

so , Such a matrix is of the form
[0 b]
[c 0] for some b, c.

Since these matrices are generated by
[0 1].[0 0]
[0 0],[1 0], the dimension equals 2. is that a right answer ?

5. Mar 28, 2012

### homeomorphic

Yes, that's it.

6. Mar 29, 2012

### Bacle2

I don't mean to use rigor, for rigor's sake, but I think it is important to note that not only
do those matrices generate, but that no smaller ( in size/cardinality) set generates the whole space.

7. Mar 29, 2012

### HallsofIvy

Staff Emeritus
Specifically, any matrix in that set can be written
$$\begin{bmatrix}0 & a \\ b & 0\end{bmatrix}= \begin{bmatrix}0 & a \\ 0 & 0 \end{bmatrix}+ \begin{bmatrix}0 & 0 \\ b & 0 \end{bmatrix}= a\begin{bmatrix}0 & 1 \\ 0 & 0\end{bmatrix}+ b\begin{bmatrix}0 & 0 \\ 0 & 1\end{bmatrix}$$
which makes it clear what a basis is.

8. Mar 31, 2012

### sakander

WHAT IS THE RESULT span(spanV)=???

9. Mar 31, 2012

### Cecilia48

Try doing a generalized row-reduction using Gaussian elimination.