MHB Dimension of $m \times n$ Matrices: Finding Basis

[Guest2]

What's the dimension of the space of $2 \times 2$ matrices? What's the dimension of the space of $m \times n$ matrices?

I know that matrices of size $m \times n$ with components in field $K$ form a vector space over $K$. To find the dimension, I would have to find basis. This I'm not quite sure how to do.

 

[I like Serena]

What's the dimension of the space of $2 \times 2$ matrices? What's the dimension of the space of $m \times n$ matrices?

I know that matrices of size $m \times n$ with components in field $K$ form a vector space over $K$. To find the dimension, I would have to find basis. This I'm not quite sure how to do.

Hi Guest,

The dimension of matrices in $\mathbb R^{2 \times 2}$ is written as $2 \times 2$, which is equal to $4$, since we need $4$ numbers to identify one.

A basis of the matrices of size $m \times n$ is the set of those matrices with a single $1 \in K$ in it and the rest $0 \in K$.
Any matrix can be written as a linear combination of those single-entry matrices.

 

[Deveno]

I will answer the first question only:

A (possible) basis for the space of 2x2 matrices is:

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

Clearly this spans, so it falls to you to show linear independence. Try generalizing this.