# Find the basis of fundamental subspaces

I was watching a video about linear algebra and computing the four fundamental subspaces. The problem that was given in the video was the following: Suppose $$B=\left[\begin{matrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ -1 & 0 & 1 \\ \end{matrix}\right] \left[\begin{matrix} 5 & 0 & 3 \\ 0 & 1 & 1 \\ 0 & 0 & 0 \\ \end{matrix}\right]$$ Find a basis for and compute the dimension of each of the 4 fundamental subspaces. Note: the matrix B is given in the B=LU form, if you have watched Gilbert Strang Lectures on Linear Algebra this form will make more sense.

They gave the solution: Dimension of column space C(B)=2 (since there are two pivots) A basis for C(B) is : $$\left[\begin{matrix} 1 \\ 2 \\ -1\end{matrix}\right] \left[\begin{matrix} 0 \\ 1 \\ 0 \\ \end{matrix}\right]$$

Dimension of row space $$C(B^T)=2$$

$$\left[\begin{matrix} 5 \\ 0 \\ 3 \\ \end{matrix}\right] \left[\begin{matrix} 0 \\ 1 \\ 1 \\ \end{matrix}\right]$$ Dimension of null space N(B) is 1 ( since there are 3 columns- number of independent columns which is 2) $$\left[\begin{matrix} -5/3 \\ -1 \\ 1 \\ \end{matrix}\right]$$ Dimension of left null space $$N(B^T)=1$$ $$\left[\begin{matrix} 1 \\ 0 \\ 1 \\ \end{matrix}\right]$$

I can't understand how they got the values for the basis, and why they use matrix L to find the basis for the column space. If someone can clarify this to me. Thank you!