# Space, basis, dimension, rank of Matrices

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• June 7th 2010, 07:59 AM
jayshizwiz
Space, basis, dimension, rank of Matrices
I'll start off easy.

How do you determine the dimension and a basis of the group of all 2x2 matrices over R. That would be $M_{2 \times 2}^R = \left\{\left(\begin{array}{cc}\alpha&\beta\\\lambd a&\delta\end{array}\right) | (\alpha, \beta, \lambda, \delta)\in R\right\}$
• June 7th 2010, 08:16 AM
tonio
Quote:

Originally Posted by jayshizwiz
I'll start off easy.

How do you determine the dimension and a basis of the group of all 2x2 matrices over R. That would be $M_{2 \times 2}^R = \left\{\left(\begin{array}{cc}\alpha&\beta\\\lambd a&\delta\end{array}\right) | (\alpha, \beta, \lambda, \delta)\in R\right\}$

Well, as it is, or should be, well known, $M_{n\times n}(\mathbb{R})\cong \mathbb{R}^{n^2}$ (isomorphism as vector spaces, among other things), so...

Tonio
• June 9th 2010, 08:59 AM
HallsofIvy
Quote:

Originally Posted by jayshizwiz
I'll start off easy.

How do you determine the dimension and a basis of the group of all 2x2 matrices over R. That would be $M_{2 \times 2}^R = \left\{\left(\begin{array}{cc}\alpha&\beta\\\lambd a&\delta\end{array}\right) | (\alpha, \beta, \lambda, \delta)\in R\right\}$

$\begin{pmatrix}\alpha & \beta \\ \gamma & \delta\end{pmatrix}$ $=\alpha \begin{pmatrix}1 & 0 \\ 0 & 0\end{pmatrix}+ \beta\begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix}+ \gamma\begin{pmatrix}0 & 0 \\ 1 & 0 \end{pmatrix}+ \delta\begin{pmatrix}0 & 0 \\ 0 & 1\end{pmatrix}$.

It's that simple!

One letter wrong!
• June 10th 2010, 03:13 AM
jayshizwiz
[QUOTE]LaTeX Error: Syntax error

It's that simple![QUOTE]

I wish it was that simple! (:
• June 10th 2010, 11:11 AM
jayshizwiz
The extra 1 after the beta is a mistake, correct?
... $\beta \left(\begin{array}{cc}0&1\\0&0\end{array}\right)$
• June 11th 2010, 09:01 AM
HallsofIvy
Yes, and thanks for catching that!

It should be
$
\begin{pmatrix}\alpha & \beta \\ \gamma & \delta\end{pmatrix}
$
$=\alpha \begin{pmatrix}1 & 0 \\ 0 & 0\end{pmatrix}+ \beta\begin{pmatrix}0 & 1 \\ 0 & 0\end{pmatrix}+ \gamma\begin{pmatrix}0 & 0 \\ 1 & 0 \end{pmatrix}+ \delta\begin{pmatrix}0 & 0 \\ 0 & 1\end{pmatrix}
$

Now, it should be easy!