I believe it was the "\\" just before \end{pmatrix} or the unecessary [ and \] that caused the (0)s. I have removed them here.

The simples thing to do, I think, is just try to choose matrices from the 5 given as generators that are independent.

For example,

is not amultipleof

so those two are independent.

Now, can

be written as a linear combination of the first two? That is, can you find numbers, a and b, such that

[tex]

\begin{pmatrix}

2 & 1 \\

1 & 9

\end{pmatrix}= a\begin{pmatrix}

1 & 2 \\

2 & 3

\end{pmatrix} + b\begin{pmatrix}

1 & -2 \\

-2 & 4

\end{pmatrix}[/MATh]?

If so, they they aredependentand you can discard the third matrix. If not, is the fourth matrix independent of the first three?