Done and Very Little Fun

So the semester is over and I am happy that it is done, but my one final was kinda a bitch to say the least. There were 10 problems total involving nullspaces and column spaces and eigenvalues and eigenvectors and all that good shit that no one really likes, unless you are a math dork like me. I was gonna put some of the questions on here and may below, so if u can solve them then feel free. So Tuesday I decided to hop back on the bike for the first time in 9 days and I felt like shit for an hour and then felt good the last hour. Problem is the reason I felt bad was I have an upper respiratory infection and maybe strep. My girlfriend has been sick and I have been around her a lot, but it is by no means her fault. Maybe I will just swing my way through it. Anyhow I think I ended the semester with three A's and one B. That final in math really fuct me in that department. Thats why I go to school though now, this degree I am there to actually learn and not just sit and hope for the best. Anyhow here are a few math questions to sink ur teeth into.

1. For what values of Y1 and Y2 would this system be consistent?
a)
{1 2 3} = Y1
{-2 -4 -5} = Y2

b) Find all the solutions to the set that make Y1 and Y2 consistent

2. Find the determinant of the following matrix

o o o 2
5 4 2 0
0 3 0 0
8 7 6 0

3. calculate the dimension of span (X1, X2, X3, X4)

2 6 4 8
6 -1 1 -1
4 4 3 11
-8 2 -1 3

4. Consider the following matrix and determine the following.
1 2 1 0 2 7
1 2 2 -1 3 12
1 2 2 -1 4 15

a) Row reduced echelon form

b) The basis for the row space and its dimension

c) The basis for the nullspace and its dimension

d) The basis for the column space and its dimension

5. Can the following be linearly independent in R4?
3 2 1
-1 1 1
4 3 1
2 -1 1

6. L: v->w is a linear transformation
Prove that the Kernel of (L) is in the subspace of V

7. L is on R2 which rotates 45 degrees counterclockwise and reflects the result about the x-axis. Give a matrix for the resulting solution based upon the standard basis e1 and e2.

8. If L is a linear operator that maps R2->R2 then solve for B if
A= 2 -8 and V= 4 2
1 -4 1 1

my hint to you is use V-1 A V. (This hint was not on the test though)

9. Find the eigen values and eigen vectors for the associated matrix.

-2 2 2
0 1 -3
0 1 5

10. Let @ be an eigenvalue of A and let x be an eigenvector belonging to @. Show that @^3 is an eigenvalue of A^3 and x is an eigenvector of A^3 belonging to @^3.

I used @ in #10 because I have no lamda on my computer here. If u have answers post them up, if not then have fun trying to solve.