# Thread: Finding a subset of S that is a basis for W.

1. ## Finding a subset of S that is a basis for W.

Let W be the subspace of M_2x2 (R) consisting of the symmetric 2x2 matrices. The set

S= {$\displaystyle [ \begin{pmatrix} 0 & -1 \\ -1 & 1 \\ \end{pmatrix} \]$, $\displaystyle [ \begin{pmatrix} 1 & 2 \\ 2 & 3 \\ \end{pmatrix} \]$$\displaystyle [ \begin{pmatrix} 2 & 1 \\ 1 & 9 \\ \end{pmatrix} \] \displaystyle [ \begin{pmatrix} 1 & -2 \\ -2 & 4 \\ \end{pmatrix} \] \displaystyle [ \begin{pmatrix} -1 & 2 \\ 2 & -1 \\ \end{pmatrix} \]} generates W. Find a subset of S that is a basis for W. (I'm not sure what all the (0)'s are next to my matrices but they aren't supposed to be there its only supposed to be the 5 matrices) 2. Originally Posted by tn11631 Let W be the subspace of M_2x2 (R) consisting of the symmetric 2x2 matrices. The set S= {\displaystyle \begin{pmatrix} 0 & -1 \\ -1 & 1 \end{pmatrix}, \displaystyle \begin{pmatrix} 1 & 2 \\ 2 & 3 \end{pmatrix}$$\displaystyle \begin{pmatrix} 2 & 1 \\ 1 & 9 \end{pmatrix}$
$\displaystyle \begin{pmatrix} 1 & -2 \\ -2 & 4 \end{pmatrix}$
$\displaystyle \begin{pmatrix} -1 & 2 \\ 2 & -1 \end{pmatrix}$}

generates W. Find a subset of S that is a basis for W.

(I'm not sure what all the (0)'s are next to my matrices but they aren't supposed to be there its only supposed to be the 5 matrices)
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,
$\displaystyle \begin{pmatrix} 1 & 2 \\ 2 & 3 \end{pmatrix}$
is not a multiple of
$\displaystyle \begin{pmatrix} 0 & -1 \\ -1 & 1 \end{pmatrix}$
so those two are independent.

Now, can
$\displaystyle \begin{pmatrix} 2 & 1 \\ 1 & 9 \end{pmatrix}$
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 are dependent and you can discard the third matrix. If not, is the fourth matrix independent of the first three?

3. It was the \]. If you include just that here you get the (0),

$\displaystyle \]$

which doesn't happen when compiling a document (or, at least, it didn't happen when I tried to).

Anyway, shouldn't it be $and$?...