# Thread: four fundamental vector spaces

1. ## four fundamental vector spaces

Compute the four fundamental vector spaces associated with A and verify the theorem "If A is a given matrix, then the null space is the orthogonal complement of the row space of A and the null space of $\displaystyle A^{T}$ is the orthogonal complement of the column space of A."

A=$\displaystyle \begin{pmatrix}1 & 5 & 3 & 7\\2 & 0 & -4 & -6\\4 & 7 & -1 & 2\end{pmatrix}$

So far, I got rref(A)=B=$\displaystyle \begin{pmatrix}1 & 0 & -2 & -3\\0 & 1 & 1 & 2\\0 & 0 & 0 & 0\end{pmatrix}$

I'm not sure if I am doing this write. The following is my work. Any advice or comments would be greatly appreciated. Thank you!

$\displaystyle x=\begin{pmatrix}2r+3s\\-r-s\\r\\s\end{pmatrix}=r\begin{pmatrix}2\\-1\\1\\0\end{pmatrix}+s\begin{pmatrix}3\\-2\\0\\1\end{pmatrix}$

$\displaystyle S={\begin{pmatrix}2\\-1\\1\\0\end{pmatrix},\begin{pmatrix}3\\-2\\0\\1\end{pmatrix}}$

T={(1, 0, -2, -3), (0, 1, 1, 2)}

and the transpose of A=$\displaystyle \begin{pmatrix}1 & 2 & 4\\5 & 0 & 7\\3 & -4 & -1\\7 & -6 & 2\end{pmatrix}$

rref(transpose of A)=$\displaystyle \begin{pmatrix}1 & 0 & {7/5}\\0 & 1 & {13/10}\\0 & 0 & 0\\0 & 0 & 0\end{pmatrix}$

$\displaystyle S'=\begin{pmatrix}0\\-1\\1\\0\end{pmatrix}, \begin{pmatrix}-7/5\\-13/10\\0\\1\end{pmatrix}$

T'=$\displaystyle \begin{pmatrix}1\\0\\7/5\end{pmatrix}, \begin{pmatrix}0\\1\\13/10\end{pmatrix}$

2. Finding your vectors that Span Nul A looks correct.