Math 215 §2, Spring 2002, Linear Algebra
Exam I

Name:

Answer all questions. 10 points per part.

1. Use the definition of linear transformation to
   1. Show that the function $f:\ensuremath{\mathbb{R}}^2\to \ensuremath{\mathbb{R}}^2$ with $f([x,y])=[x^2-y^2,x+y-1]$ is not a linear transformation.

   2. Show that the function $g:\ensuremath{\mathbb{R}}^2\to \ensuremath{\mathbb{R}}^2$ with $g([x,y])=[x+y,2x+4y]$ is a linear transformation.

2. Find the inverse of the matrix
   
   $$\left( \begin{array}{cc} 2 & 4 \\ 3 & 7 \end{array}\right)$$
   
   

3. Using a basis:
   1. Prove that $B=([1,2],[-1,1])$ is an ordered basis for $\ensuremath{\mathbb{R}}^2$.

   2. Give the B-coordinates for $[1,5]$

   3. Give the matrix for $g$ with respect to this ordered basis (as in 1b, we have $g([x,y])=[x+y,2x+4y]$).

   4. Use that matrix to find the B-coordinates for $g([1,5])$.

4. What are $\mbox{Ker}(g)$ and $\mbox{Im}(g)$?

5. What are the eigenvalues of $g$?

6. What is the vector in the direction of $[1,4]$ closest to $[3,-2]$?