# Basis

• Feb 8th 2010, 07:40 PM
millerst
Basis
Exhibit a basis for the given space and prove that it is a basis.

The space of 2 x 2 matrices.

I don't understand how to this at all, can someone take me through this example step by step and kind of explain why/concepts involved.

Thanks so much.
• Feb 9th 2010, 12:28 AM
Roam
Quote:

Originally Posted by millerst
Exhibit a basis for the given space and prove that it is a basis.

The space of 2 x 2 matrices.

I don't understand how to this at all, can someone take me through this example step by step and kind of explain why/concepts involved.

Thanks so much.

It's the 2x2 identity matrix, since it is linearly independent and spanning.
• Feb 9th 2010, 02:59 AM
HallsofIvy
I can't understand what Roam said at all since the words "independent" and "spanning" apply to sets of vectors, not individual vectors or matrices.

millerst, you probably know that a basis for, say, $R^4$, the set of quadruples, (a, b, c, d), is {(1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1)} since we can write (a, b, c, d)= (a, 0, 0, 0)+ (0, b, 0, 0)+ (0, 0, c, 0)+ (0, 0, 0, d)= a(1, 0, 0, 0)+ b(0, 1, 0, 0)+ c(0, 0, 1, 0)+ d(0, 0, 0, 1).

Okay, any matrix in your space can be written as
$\begin{bmatrix}a & b \\ c & d\end{bmatrix}$

You want matrices M1, M2, M3, M4 so that
$\begin{bmatrix}a & b \\ c & d\end{bmatrix}= aM1+ bM2+ cM3+ dM4$
there are obvious matrices that do that.
• Feb 9th 2010, 09:57 AM
Roam
The standard basis for $M_{22}$ consists of the following four matrices:

$\left\{ \begin{bmatrix}1 & 0 \\ 0 & 0\end{bmatrix}, \begin{bmatrix}0 & 1 \\ 0 & 0\end{bmatrix}, \begin{bmatrix}0 & 0 \\ 1 & 0\end{bmatrix}, \begin{bmatrix}0 & 0 \\ 0 & 1\end{bmatrix} \right\}$

Because the space of 2x2 matrices has dimension 4, so can't be spanned by just two vectors.

Sorry for my carelessness...