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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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    @dexter04 If a basis for [the set of all 2x2 matrices of real numbers] is in the span of these four matrices, then does the set of four matrices span [the set of all 2x2 matrices of real numbers]?2012-12-12

2 Answers 2

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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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    @Makogan, a million questions have been posted to m.se without incident, but, suit yourself.2017-10-22
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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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    Yes, my mistake.2012-12-12