Linear Algebra Review (for Math 5595, Fall 1999)

Instructions: This review is intended as a self-test to determine if your linear algebra skills are sufficient for this course. Look over each problem and determine whether or not you know how to do the problem (you don't actually have to work the problem). For each problem which you don't know how to do, go back to your linear algebra text and review the appropriate material so that you can work all of the following problems. If you are not able to solve all of these problems with the aid of a linear algebra text, you are not ready for this course.
1. For which value of k does the following system of linear equations have no solutions? Exactly one solution? Infinitely many solutions?
```    x-y=3
2x-2y=k
```
2. Solve the following system of equations by Gaussian elimination
```     x+y+2z=8
-x-2y+3z=1
3x-7y+4z=10
```
3. Compute AB, given that
```A=[ 1 2 3]    B=[1 2 3]
[ 2 2 2]      [4 5 6]
[7 8 9]
```
4. Compute the Euclidean norm of v=(1,3,9).
5. Find the inverse of
```        [1 2 3]
A =  [2 5 3]
[1 0 8]
```
6. Given two vectors v,w in R^n; what can be said about the angle between v and w if their dot product is negative? zero? positive?
7. Give an equation for the plane in R^3 which passes through the point (1,2,3) and is orthogonal to the vector (2,-1,-1).
8. Express the vector (2,2,2) as a linear combination of u=(0,-2,2) and v=(1,3,-1).
9. Find an equation for the plane spanned by the vectors u=(-1,1,1,) and v=(3,4,4).
10. For which real values of k do the following vectors form a linearly dependent set in R^3?
```u=(k,-1,-1), v=(-1,k,-1), w=(-1,-1,k)
```
11. Determine a bases for the plane 3x-2y+5z=0 in R^3.
12. Find a bases for the nullspace of
```     [1 -1  3]
A = [5 -4 -4]
[7 -6  2]
```
13. Determine the rank of the matrix A above.