Its for a calculus homework. They give me 2 subspace basis and a matrix, and after row reduction it appears the sum and the intersection. I have to explain why and how appears the intersection. This is the homework.
$ basis-S = {(1,2,1,1,1),(1,0,1,0,1),(-1,1,0,1,1)} $
$ basis-W = {(1,1,1,2,-2),(1,3,1,3,-2)} $
So, in the matrix we have:
- Green -> basis-S
- Yellow -> basis-S
- Orange -> basis-W
- Purple -> null, 0
After the row reduction, we have:
- Sum -> $ (1,2,1,1,1)(0,1,0,2,-3)(0,0,-1,4,-11)(0,0,0,3,-6) $ (upper left)
- Intersection -> $ (0,2,0,1,0) $ (bottom right)
So i have to explain why appears the intersection in the right. How the information of basis-W pass to the right and appears the intersection.
If you can help me, i appreciate it. Any question, tell me. Thanks!
I'll use $B_S$ and $B_W$ to denote the matrices containing the given bases, so the initial matrix can be written in block form as $$ \begin{bmatrix}B_S&B_S\\B_W&0\end{bmatrix}. $$ Hint: An element of the row space of the original matrix is of the form $$ \begin{bmatrix}a&b\end{bmatrix} \begin{bmatrix}B_S&B_S\\B_W&0\end{bmatrix} =\begin{bmatrix}aB_S+bB_W&aB_S\end{bmatrix} $$ If the first half of this vector is zero, what does it tell you? Now recall that the row reduced matrix has the same row space as the original matrix.