Question

subspace w is defined by: x1 - x2 + x3 = 0 and x1 - x4 =0; v is the orthogonal complement; what is the image of matrix v?

Answer 1

I can take the first equation and treat it as a column vector so become 1 , -1, 1, 0, and dot it with x, and it's going to be 0. Same with the other one 1,0,0,-1.
Any vector that's in V must be orthogonal to both of these vectors so I can write this like a matrix equation, and we do this by combining these two vectors as rows of the matrix.
This matrix it's already row reduced, you must only reverse the two row.
\[\left(\begin{matrix}1 & 0 & 0 & -1 \\ 1 & -1 & 1 & 0\ \end{matrix}\right) x\]
We see here that this matrix has a rank of 2. Now, we're looking at vectors which live in R4, so we know that the null space is going to have a dimension which is 4 minus 2.
To parameterize these two dimensional vectors, what I'm going to do is I'm going to let x3 equals some constant, and x2 equal another constant. So specifically, I'm going to let x3 equal b, and x2 equal a.
\[\left(\begin{matrix}x1 \\ x2\\ x3\\x4\end{matrix}\right) =\left(\begin{matrix}a-b \\ a\\b\\a-b\end{matrix}\right)\]And now what we can do is we can take this vector and we can decompose it into pieces which are a multiplied by a vector, and b multiplied by a vector.
\[a \left(\begin{matrix}1 \\ 1\\ 0\\-1\end{matrix}\right) + b \left(\begin{matrix}-1 \\ 0\\1\\-1\end{matrix}\right)\]
So V is going to be spanned by this vector and this vector.