# Space, basis, dimension, rank of Matrices

• Jun 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 $\displaystyle M_{2 \times 2}^R = \left\{\left(\begin{array}{cc}\alpha&\beta\\\lambd a&\delta\end{array}\right) | (\alpha, \beta, \lambda, \delta)\in R\right\}$
• Jun 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 $\displaystyle M_{2 \times 2}^R = \left\{\left(\begin{array}{cc}\alpha&\beta\\\lambd a&\delta\end{array}\right) | (\alpha, \beta, \lambda, \delta)\in R\right\}$

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

Tonio
• Jun 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 $\displaystyle M_{2 \times 2}^R = \left\{\left(\begin{array}{cc}\alpha&\beta\\\lambd a&\delta\end{array}\right) | (\alpha, \beta, \lambda, \delta)\in R\right\}$

$\displaystyle \begin{pmatrix}\alpha & \beta \\ \gamma & \delta\end{pmatrix}$$\displaystyle =\alpha \begin{pmatrix}1 & 0 \\ 0 & 0\end{pmatrix}+ \beta\begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix}+ \gamma\begin{pmatrix}0 & 0 \\ 1 & 0 \end{pmatrix}+ \delta\begin{pmatrix}0 & 0 \\ 0 & 1\end{pmatrix}. It's that simple! One letter wrong! • Jun 10th 2010, 03:13 AM jayshizwiz [QUOTE]LaTeX Error: Syntax error It's that simple![QUOTE] I wish it was that simple! (: • Jun 10th 2010, 11:11 AM jayshizwiz The extra 1 after the beta is a mistake, correct? ... \displaystyle \beta \left(\begin{array}{cc}0&1\\0&0\end{array}\right) • Jun 11th 2010, 09:01 AM HallsofIvy Yes, and thanks for catching that! It should be \displaystyle \begin{pmatrix}\alpha & \beta \\ \gamma & \delta\end{pmatrix}$$\displaystyle =\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!