6.1#8

Hello professor Taylor,

I was wondering about how I would go about finding the basis for the eigenvector for the 位1 value of -1. I understand the way you find it when the 位 is zero, as it is simply gotten by putting matrix A in RREF form and then solving for the null space of it, however, I am not sure how to work this out for the 位 value of -1.
The first thing I did was find the eigenvalues by taking the determinate of the matrix after adding in the -位 values to it. This yielded 位^4+2位^3+位^2. After solving this system of equations I got 位=0 (with multiplicity 2 because it was 位^2=0) and 位=-1 (with multiplicity 2 because it was (位+1)^2.
For the eigenspace of -1 I calculated:
-----------------
|1,   0, -1, -1|
|-1,  0,  0,  0|
|-2,  0,  2,  2|
|2,   0, -1, -1|
-----------------
Then, putting this in RREF I got
---------------
|1, 0,  0,  0|
|0, 0,  1,  1|
|0, 0,  0,  0|
|0, 0,  0,  0|
---------------
This would mean that the null space is:
-----
| 0|
| 0|
|-1|
| 1|
-----
However, this wasn't the basis which the question was asking for. Where did I mess up? Thank you in advance for your help!
**********************************
Some of your math isn't rendering properly, so I have to guess a little bit.
First of all, the null space is a space and not a vector. You could hope that the vector you wrote is a basis for the null space, but it turns out this is not the case. You haven't done completely badly however, in fact you've done everything properly except interpret the meaning of RREF(A-(-1)I).  When you solve the equation
---------------  ----      ---
|1, 0,  0,  0|    |x1|      |0|
|0, 0,  1,  1|    |x2|   = |0|
|0, 0,  0,  0|    |x3|      |0|
|0, 0,  0,  0|    |x4|      |0|
---------------  ---       ---
you get the equations x1=0,  and x3+x4 = 0. You deduce correctly that x3 must equal -x4, but you've missed the fact in all of this that x2 is not constrained at all!  Thus all vectors of the form
____            ____             _____
|  0  |            |   0   |            |   0   |
| x2 |            |   1   |            |   0   |
|-x4 |  =   x2 |   0  |   +  x4 |  -1   |
| x4 |            |   0  |             |  -1   |
-----             ------              -------
So you get two basis vectors for that eigenspace, the one you computed and in addition you need [0,1,0,0].