###### Exercise10

Consider the matrix

\begin{equation*} A = \left[\begin{array}{rrrr} -1 \amp 1 \amp 0 \amp 0 \\ 1 \amp -2 \amp 1 \amp 0 \\ 0 \amp 1 \amp -2 \amp 1 \\ 0 \amp 0 \amp -1 \amp 1 \\ \end{array}\right]\text{.} \end{equation*}
1. Find the $$LU$$ factorization of $$A\text{.}$$

2. Use the factorization to find a basis for $$\nul(A)\text{.}$$

3. We have seen that $$\nul(A) = \nul(U)\text{.}$$ Is it true that $$\col(A) = \col(L)\text{?}$$

in-context