Problem: Let V be the subspace of all 2×2 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?
Best Answer
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 $M_4=\frac{4}{3} M_1-\frac{1}{3}M_2$