Find basis and dimension of a subspace
Problem: Let V be the subspace of all 2x2 matrices over R, and W the subspace spanned by:
\begin{bmatrix} 1 & -5 \\ -4 & 2 \\ \end{bmatrix} \begin{bmatrix} 1 & 1 \\ -1 & 5 \\ \end{bmatrix} \begin{bmatrix} 2 & -4 \\ -5 & 7 \\ \end{bmatrix} \begin{bmatrix} 1 & -7 \\ -5 & 1 \\ \end{bmatrix}
Q: Find a basis and the dimension of W.
What I've done so far:
-I'll refer to the matrices as W1, W2, W3 and W4 (top-down). I've noticed so far that W3 = W1 + W2, does that means that span(W1,W2,W3,W4) = span(W1,W2,W3) ?
I know that a way to find a basis is by reducing a matrix of coefficients to a echelon form, but how do I represent those matrices in a coefficients matrix?
$\endgroup$2 Answers
$\begingroup$You can consider each matrix to be a vector in $\mathbb{R}^4$.
The only pivots are in the first two columns, so the first two matrices are linearly independent and form a basis for the subspace. The last two are linear combinations of the first. Notice that $M_3=M_1+M_2$, and $M_4=\frac{4}{3} M_1-\frac{1}{3}M_2$
$\endgroup$ 1 $\begingroup$To solve your problem, you should look for the maximum set of linearly independent matrices.
For example, if $M_1$,$ M_2$, and $M_4$ are linearly independent, but $M_1$, $M_2$, $M_3$ and $M_4$ are not, then the maximum set of linearly independent matrices are $M_1$, $M_2$ and $M_4$, then this set is a basis of the subspace W. So now you have a basis, just count the elements and you have the dimension.
I hope this may help.
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