###### Exercise 10

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*}

Find the \(LU\) factorization of \(A\text{.}\)

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

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