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Show that the vector space $M_2$ has a basis $$e_{11} = \begin{pmatrix} 1 & 0 \\ 0 & 0 \\ \end{pmatrix},\quad e_{12} = \begin{pmatrix} 0 & 1 \\ 0 & 0 \\ \end{pmatrix} ,\quad e_{21} = \begin{pmatrix} 0 & 0 \\ 1 & 0 \\ \end{pmatrix},\quad e_{22} = \begin{pmatrix} 0 & 0 \\ 0 & 1 \\ \end{pmatrix} $$ by doing the following steps:

$a.$ Show that the matrices $e_{11}, e_{12}, e_{21}, e_{22}$ span $M_2$.
$b.$ Show that the matrices $e_{11}, e_{12}, e_{21}, e_{22}$ are linearly independent.

I'm not sure how you would find the span and linear independence of matrices.

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  • $\begingroup$ Do you know what does it mean to say a set is linearly independent? $\endgroup$
    – la flaca
    Oct 19, 2016 at 0:08
  • $\begingroup$ I think it means that the 0 vector can be written as part of the set. $\endgroup$
    – user380174
    Oct 19, 2016 at 0:12
  • $\begingroup$ Not exactly, I'll post an answer guiding you $\endgroup$
    – la flaca
    Oct 19, 2016 at 0:18

2 Answers 2

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a) Consider a $2\times 2 $ matrix. Can you write it as a linear combination of those four matrices? Can you do this for any $2 \times 2$ matrix? If so, then they span $M_2$; otherwise they do not span.

$\begin{bmatrix}a&b\\c&d\end{bmatrix} = ae_{11} + be_{12} + ce_{21} + de_{22}$

b) Is there a way to write the zero matrix as a linear combination of the four matrices (excluding the trivial linear combination $0 e_{11} + 0e_{12} + 0e_{21} + 0e_{22}$)? If yes, they are linearly dependent; otherwise, they are linearly independent.

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  • $\begingroup$ So for (a), would I pick a random 2 x 2 matrix to see if it's spans M2 or would I pick an arbitrary one like with a, b, c, d. $\endgroup$
    – user380174
    Oct 19, 2016 at 0:30
  • $\begingroup$ For (b), would I have to do this for all the matrices? $\endgroup$
    – user380174
    Oct 19, 2016 at 0:31
  • $\begingroup$ @sara For (a) it has to work for all matrices, so you should show it for an arbitrary matrix a,b,c,d as you suggested. For (b), I made a mistake; I will rewrite it. $\endgroup$
    – angryavian
    Oct 19, 2016 at 0:37
  • $\begingroup$ for a, I'm not sure how to write it as a linear combination. I tried to make a equation equaling to the arbitrary matrix but I'm not sure how to solve it. $\endgroup$
    – user380174
    Oct 19, 2016 at 1:31
  • $\begingroup$ @sara See my edit. $\endgroup$
    – angryavian
    Oct 19, 2016 at 1:32
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$\{e_{11},e_{12},e_{21},e_{22}\}$ is LI iff for the only linear combination of that set that equals zero is the one with all coefficients equal to zero. So to prove that set is LI you must prove that if $a_{11}e_{11}+a_{12}e_{12}+a_{21}e_{21}+a_{22}e_{22}=0$, then all those coefficients $a_{ij}$ are zero.

$\{e_{11},e_{12},e_{21},e_{22}\}$ spans $M_2$ if you can express any $2×2$ matrix as a linear combination of that set.

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