# Math Help - 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= { $[
\begin{pmatrix}
0 & -1 \\
-1 & 1 \\
\end{pmatrix} \]$
, $[
\begin{pmatrix}
1 & 2 \\
2 & 3 \\
\end{pmatrix} \]$
$[
\begin{pmatrix}
2 & 1 \\
1 & 9 \\
\end{pmatrix} \]$

$[
\begin{pmatrix}
1 & -2 \\
-2 & 4 \\
\end{pmatrix} \]$

$[
\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= { $
\begin{pmatrix}
0 & -1 \\
-1 & 1
\end{pmatrix}$
, $
\begin{pmatrix}
1 & 2 \\
2 & 3
\end{pmatrix}$
$
\begin{pmatrix}
2 & 1 \\
1 & 9
\end{pmatrix}$

$
\begin{pmatrix}
1 & -2 \\
-2 & 4
\end{pmatrix}$

$
\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,
$
\begin{pmatrix}
1 & 2 \\
2 & 3
\end{pmatrix}$

is not a multiple of
$
\begin{pmatrix}
0 & -1 \\
-1 & 1
\end{pmatrix}$

so those two are independent.

Now, can
$
\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),

$\]$

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

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