# 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]?

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I need to find a more concise notation for "the set of all 2x2 matrices" - the question above seems too verbose. – Anderson Green Dec 12 '12 at 1:30
If you know the underlying ring, use matrix ring. For invertible matrices, use general linear group - $GL_n(\mathbb R)$ – dexter04 Dec 12 '12 at 1:39
The standard definition of rings in abstract algebra - groups, rings, fields etc. – dexter04 Dec 12 '12 at 1:44
The set of all 2x2 matrices is usually denoted by $M_2(\mathbb{R})$ or $\mathbb{R}^{2\times 2}$. – user1551 Dec 12 '12 at 1:49
@dexter04 Is there any way to find the underlying ring for the set of four matrices here? – Anderson Green Dec 12 '12 at 1:51

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 $a,b,c,d$)?

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This looks more like a question than an answer - does that make it a rhetorical answer? – Anderson Green Dec 12 '12 at 2:51
It makes it an opportunity for you to have the joy of working out the details on your own. – Gerry Myerson Dec 12 '12 at 3:06
I think the formulas would be a = r, b = s, c = t, and d = (r + u). In this case, it is clear that the matrices span the set of all 2x2 matrices. Does that mean that the four matrices are a basis for the set of all 2x2 matrices? – Anderson Green Dec 12 '12 at 17:02
You have written the formulas for $a,b,c,d$ in terms of $r,s,t,u$. What you need is formulas for $r,s,t,u$ in terms of $a,b,c,d$, since you are trying to show that every matrix ["for every $a,b,c,d$"] is a linear combination of the four matrices ["there exist $r,s,t,u$"]. – Gerry Myerson Dec 12 '12 at 22:31

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.

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Of course, the zero matrix is also a linear combination of the first and last matrices in the standard basis, so one needs to say a little more. – Gerry Myerson Dec 12 '12 at 2:20
@ruwin Did you mean to write "all 2x2 matrices are in the span of the four given matrices." instead? – Anderson Green Dec 12 '12 at 2:45
Yes, my mistake. – ruwin Dec 12 '12 at 3:08