What is the Dimension of the Subspace of M2;2 with Zero Diagonal Entries?

[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 ,
thanks in advance

 

[Bacle2]

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

 

[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.

 

[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 ?

 

[homeomorphic]

Yes, that's it.

 

[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.

 

[HallsofIvy]

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.

 

[sakander]

WHAT IS THE RESULT span(spanV)=?

 

[Cecilia48]

Try doing a generalized row-reduction using Gaussian elimination.