Dimension of the linear space $\mathbb{R}^{n\times n}$?

Question

i) What is the dimension of the linear space $\mathbb{R}^{n\times n}$ of all $n\times n$ matrices?

ii) What is the dimension of the subspace of symmetric matrices in $\mathbb{R}^{n\times n}$?

Def: The dimension of the linear space $V$ is defined as the maximum number of linearly independent vectors in $V$.

I really don't understand att all how I can use that definition to answer i) and ii) above. How should I think through this?

Answer 1

HINT

Think to a possible standard basis for the two subspaces.

Case 2-by-2 - General case

$$\begin{bmatrix}a&b\\c&d\end{bmatrix}=av_1+bv_2+cv_3+dv_4=a\begin{bmatrix}1&0\\0&0\end{bmatrix}+b\begin{bmatrix}0&1\\0&0\end{bmatrix}+c\begin{bmatrix}0&0\\1&0\end{bmatrix}+d\begin{bmatrix}0&0\\0&1\end{bmatrix}$$

Case 2-by-2 - Symmetric

$$\begin{bmatrix}a&b\\b&c\end{bmatrix}=aw_1+bw_2+dw_3=a\begin{bmatrix}1&0\\0&0\end{bmatrix}+b\begin{bmatrix}0&1\\1&0\end{bmatrix}+c\begin{bmatrix}0&0\\0&1\end{bmatrix}$$

Answer 2

There are very basic $n\times n$ matrices consisting of a $1$ in position $ij$ and zeroes on the rest of the entries. Those allow you to write a convenient linear combinations for any generic $n\times n$ matrix.