Matrix Spaces

$3$ by $3$ matrices §

We identified $M$, the space of all $3$ by $3$ matrices; S, the space of all symmetric $3$ by $3$ matrices; U, the space of all upper triangular $3$ by $3$ matrices and D, the space of all diagonal $3$ by $3$ matrices.

You can see the introduction of Matrix Type

The dimension of $M$ is $9$. A good choice of basis is:

$$\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right],\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right],\left[\begin{array}{ccc}0& 0& 1\\ 0& 0& 0\\ 0& 0& 0\end{array}\right],\dots ,\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 1& 0\end{array}\right],\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 1\end{array}\right].$$

The dimension of $S$ is 6 and one basis is:

$$\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right],\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 0\end{array}\right],\left[\begin{array}{ccc}0& 0& 1\\ 0& 0& 0\\ 1& 0& 0\end{array}\right],\left[\begin{array}{ccc}0& 0& 0\\ 0& 1& 0\\ 0& 0& 0\end{array}\right],\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right],\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 1\end{array}\right].$$

The dimension of $U$ is also 6 and a basis for $U$ is:

$$\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right],\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right],\left[\begin{array}{ccc}0& 0& 1\\ 0& 0& 0\\ 0& 0& 0\end{array}\right],\left[\begin{array}{ccc}0& 0& 0\\ 0& 1& 0\\ 0& 0& 0\end{array}\right],\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 1\\ 0& 0& 0\end{array}\right],\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 1\end{array}\right].$$

The subspace $D=S\cap U$ has dimension 3. A good basis is:

$$\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right],\left[\begin{array}{ccc}0& 0& 0\\ 0& 1& 0\\ 0& 0& 0\end{array}\right],\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 1\end{array}\right].$$

$S\cup U$ is NOT a Subspaces of $M$. However, $S+U$, which means all possible sums of elements of $S$ and elements of $U$, is a subspace of $M$. In fact, the subspace is just $M$ itself. We can find that dimensions follow this rule:

$$\text{dim }S+\text{dim }U=\text{dim }(S+U)+\text{dim }(S\cap U).$$

Differential equations §

We can think of the solutions $y$ to $\frac{{\mathrm{d}}^{2}y}{\mathrm{d}{x}^2}+y=0$ as the elements of a Nullspace.

The complete solution is:

$$y=c_1\cos x+c_2\sin x,$$

where $c_1$ and $c_2$ can be any complex numbers. The solution space is a two dimensional vector space with Basis vectors $\cos x$ and $\sin x$.