Homework #2

1. Find \( A \) and \( B \) where...
   \( A = \begin{bmatrix} 1 & ? & 1 & ? & ? & 3 & ? & ? \\ 0 & ? & 2 & ? & ? & 1 & ? & ? \\ 2 & ? & 2 & ? & ? & -1 & ? & ? \\ -1 & ? & 1 & ? & ? & 2 & ? & ? \end{bmatrix} \quad \stackrel{\mathrm{RREF}}{\sim} \quad \begin{bmatrix} 1 & 3 & 0 & -1 & -1 & 0 & 0 & 2 \\ 0 & 0 & 1 & 2 & 1 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{bmatrix} \)             \( B = \begin{bmatrix} ? & ? & -2 & ? & -2 & ? \\ ? & ? & -4 & ? & 5 & ? \\ ? & ? & -6 & ? & 7 & ? \\ ? & ? & -2 & ? & 0 & ? \end{bmatrix} \quad \stackrel{\mathrm{RREF}}{\sim} \quad \begin{bmatrix} 0 & 1 & -2 & 0 & 3 & 3 \\ 0 & 0 & 0 & 1 & -1 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{bmatrix} \)
   
   
2. Define \( T:\mathbb{R}^{2 \times 2} \to \mathbb{R}[x] \) as follows: \( T \left( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \right) = (a+2b+3d)x^2+(-a-2b+c+d)x+(a+2b+c+7d) \)
   Note that \( T \) is a linear transformation (you don't need to prove this).
   1. Write down the standard coordinate matrix for \( T \) and find its RREF [You may truncate the infinitely many rows of zeros.]
   2. Find a basis for the kernel and range of \( T \).
   3. What is the nullity and rank of \( T \)? Is \( T \) 1-1, onto, both, neither?
   
   
3. Consider \( W = \left\{ \begin{bmatrix} -3a-4b+c & b-c & -4a-6b+2c \\ a+b & 2b-2c & a+b \end{bmatrix} \;\Big|\; a,b,c \in \mathbb{R} \right\} \) and \( V = \left\{ \begin{bmatrix} a & b & c \\ d & e & f \end{bmatrix} \;\Bigg|\; \begin{array}{ccc} a+3b+2c+4d+e+7f & = & 0 \\ 2a+6b+c+5d-e+5f & = & 0 \\ 3a+9b+c+7d-2e+6f & = & 0 \end{array} \right\} \).
   1. Both \( W \) and \( V \) are subspaces of \( \mathbb{R}^{2 \times 3} \). Show this is the case by re-expressing each as either a span or kernel [thus it's a subspace since spans and kernels are always subspaces]. Pick the most convenient description.
   2. Show \( W \subseteq V \).
   3. Find a basis for \( V \).
   4. Find a basis for \( W \). Then extend this to a basis for all of \( V \).
   
   
4. Curtis page 35 exercise #3.
   
   
5. Curtis page 45 exercise #3.
   
   
6. Curtis page 46 exercise #10 [See page 45 exercise #6].