6.7 Matrix subspaces, basis, and dimension

So far, most basis and dimension examples have used column vectors. But the same ideas apply to matrices because matrices can be added and scaled. A matrix can be a "vector" inside a vector space whose elements are matrices.

This note explains how to read basis and dimension for subspaces of $M_{m,n}(R)$, the vector space of all real $m \times n$ matrices.

Matrices as vectors

The operations are ordinary matrix addition and scalar multiplication:

$$A+B,\qquad \alpha A.$$

These operations keep the size fixed. If $A$ and $B$ are both $m \times n$, then every linear combination

$$\alpha A+\beta B$$

is again an $m \times n$ matrix.

This is a subspace of $M_{m,n}(R)$ for the same reason that the span of vectors is a subspace: sums and scalar multiples of linear combinations remain linear combinations.

The standard matrix units

For each position (i,j), define $E_{ij}$ to be the matrix with 1 in the (i,j) position and 0 everywhere else.

For example, a basis of $M_{2,2}(R)$ is

$$E_{11}= \begin{bmatrix}1&0\\0&0\end{bmatrix}, \quad E_{12}= \begin{bmatrix}0&1\\0&0\end{bmatrix}, \quad E_{21}= \begin{bmatrix}0&0\\1&0\end{bmatrix}, \quad E_{22}= \begin{bmatrix}0&0\\0&1\end{bmatrix}.$$

Every $2 \times 2$ matrix decomposes uniquely as

$$\begin{bmatrix}a&b\\c&d\end{bmatrix} =aE_{11}+bE_{12}+cE_{21}+dE_{22}.$$

The number mn is not mysterious. It is the number of independent entry positions in an $m \times n$ matrix.

Upper triangular matrices

Some matrix subspaces are defined by forcing certain entries to be zero.

Only entries on or above the diagonal are free. Therefore a basis is given by the matrix units $E_{ij}$ with $i\le j$.

For $UT_3(R)$, a basis is

$$E_{11},E_{12},E_{13},E_{22},E_{23},E_{33}.$$

Thus

$$\dim UT_3(R)=6.$$

In general,

$$\dim UT_n(R)=1+2+\cdots+n=\frac{n(n+1)}2.$$

Skew-symmetric matrices

Another important subspace is defined by an equation involving transpose.

The diagonal entries of a skew-symmetric matrix must be zero, because $a_{ii}=-a_{ii}$ implies $a_{ii}=0$. Entries above the diagonal determine the entries below the diagonal with opposite sign.

For $n=3$, every skew-symmetric matrix has the form

$$\begin{bmatrix} 0&a&b\\ -a&0&c\\ -b&-c&0 \end{bmatrix}.$$

So a basis of $Skew_3(R)$ is

$$E_{12}-E_{21},\qquad E_{13}-E_{31},\qquad E_{23}-E_{32},$$

and

$$\dim Skew_3(R)=3.$$

In general,

$$\dim Skew_n(R)=\frac{n(n-1)}2.$$

A dictionary between matrices and long column vectors

One reason matrix subspaces behave like vector subspaces is that a matrix can be turned into a long column vector by stacking its columns. For example,

$$\begin{bmatrix} a&b\\ c&d \end{bmatrix} \quad\longleftrightarrow\quad \begin{bmatrix} a\\c\\b\\d \end{bmatrix}.$$

This conversion is only a bookkeeping device, but it explains why the same linear-combination, independence, basis, and dimension theorems transfer to matrix spaces. Linear relations among matrices become linear relations among their stacked coordinate vectors.

Worked example: symmetric $2 \times 2$ matrices

Let $Sym_2(R)$ be the set of symmetric $2 \times 2$ matrices. A general element has the form

$$\begin{bmatrix} a&b\\ b&c \end{bmatrix}.$$

Therefore

$$\begin{bmatrix} a&b\\ b&c \end{bmatrix} = a\begin{bmatrix}1&0\\0&0\end{bmatrix} +b\begin{bmatrix}0&1\\1&0\end{bmatrix} +c\begin{bmatrix}0&0\\0&1\end{bmatrix}.$$

The three matrices

$$\begin{bmatrix}1&0\\0&0\end{bmatrix}, \qquad \begin{bmatrix}0&1\\1&0\end{bmatrix}, \qquad \begin{bmatrix}0&0\\0&1\end{bmatrix}$$

span $Sym_2(R)$.

They are linearly independent because

$$\alpha\begin{bmatrix}1&0\\0&0\end{bmatrix} +\beta\begin{bmatrix}0&1\\1&0\end{bmatrix} +\gamma\begin{bmatrix}0&0\\0&1\end{bmatrix} = \begin{bmatrix}0&0\\0&0\end{bmatrix}$$

forces $\alpha=0$, $\beta=0$, and $\gamma=0$ by comparing entries. Thus they form a basis and

$$\dim Sym_2(R)=3.$$

Problem-solving routine

When a problem asks for a basis of a matrix subspace:

1. write a general matrix in the subspace;
2. identify the free parameters;
3. split the matrix into one parameter times one fixed matrix, plus another parameter times another fixed matrix, and so on;
4. use the fixed matrices as a candidate basis;
5. check independence by comparing entries.

This is the matrix-space version of parameterizing a solution set and reading off the direction vectors.

Common mistakes

Quick checks

Exercises

Read this first

This note uses 6.5 Basis and dimension and 3.2 Transpose and special matrices.