Determine whether a set of matrices spans another set of matrices I'm trying to determine whether [the set of all 2x2 matrices] is in the span of the following matrices:
1 0
0 1
0 1
0 0
0 0
1 0
0 0
0 1
If a basis for [the set of all 2x2 matrices] is in the span of these four matrices, then does the set of matrices span [the set of all 2x2 matrices]? Also, is there a faster way to determine whether the set spans [the set of all 2x2 matrices]?
 A: You are being asked whether it is true that every $2\times2$ matrix is a linear combination of the four matrices you are given. That is, you are being asked whether it is true that no matter what $a,b,c,d$ are you can find $r,s,t,u$ such that $$\pmatrix{a&b\cr c&d\cr}=r\pmatrix{1&0\cr0&1\cr}+s\pmatrix{0&1\cr0&0\cr}+t\pmatrix{0&0\cr1&0\cr}+u\pmatrix{0&0\cr0&1\cr}$$ When it's written that way, can you decide whether such $r,s,t,u$ exist? Can you, in fact, go even farther and find formulas for $r,s,t,u$ (in terms of $a,b,c,d$)?
A: The standard basis for all 2x2 matrices is:
$$
        \begin{matrix}
        1 & 0 \\
        0 & 0 \\
        \end{matrix}
$$
$$
        \begin{matrix}
        0 & 1 \\
        0 & 0 \\
        \end{matrix}
$$$$
        \begin{matrix}
        0 & 0 \\
        1 & 0 \\
        \end{matrix}
$$$$
        \begin{matrix}
        0 & 0 \\
        0 & 1 \\
        \end{matrix}
$$
The first matrix in your problem
$$
        \begin{matrix}
        1 & 0 \\
        0 & 1 \\
        \end{matrix}
$$
is a linear combination of the the first and last matrices in the basis. So yes, the 4 given matrices are in the span of all 2x2 matrices.
