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2. (75 pts) Given the matrix $A = \begin{bmatrix} 4 & 3 & 0 \\ -7 & -6 & -1 \\ 1 & 3 & 3 \end{bmatrix}$.
a) (10 pts) Find a basis for the column space of matrix A. Show your work.
b) (5 pts) Do the columns of matrix A form a basis for the space $\mathbb{R}^4$? Justify.
c) (10 pts) Find a basis for the row space of matrix A. Show your work.
d) (5 pts) What is the dimension of the row space of matrix A? Justify.
e) (15 pts) Find a basis for the null space of A. Show your work. What is the nullity of A (i.e., the dimension of the null space of A)?
f) (15 pts) Find a basis for the left null space of A. Show your work. What is the dimension of this space?
g) (10 pts) Consider the system $Ax = b$. Is it possible to find a solution vector $x$ for all $b \in \mathbb{R}^4$? Discuss. Are there any vectors $b$ for which there is at least one solution?
h) (5 pts) Find det(A)? Show your work or otherwise justify your answer.
