# Finding rank(range) and nullspace of a matrix

1. Jun 13, 2012

### caliboy

1. The problem statement, all variables and given/known data
Trying to figure out the rank and nullspace of the matrix of matrix A and B:

A=
1 0
5 4
1 4

B=
1 0 1
5 4 9
2 4 6
2. Relevant equations
I used the Guass elimination on both

3. The attempt at a solution

For A I said r3$\rightarrow$r3-r1, then r3→r3+4r1 then r2→r2-5r1 that lead to me getting
A=
1 0
0 4 Rank=2 and Null space=0
0 0

For B I said r3→r3-r2, then r3→r3+3r1 that lead me to:
B=
1 0 1
5 4 9 Rank=2 and Null space=1
0 0 0

Am I on the right track or do I have these completly wrong?

2. Jun 13, 2012

### Simon Bridge

3. Jun 13, 2012

### vela

Staff Emeritus
Are you trying to find the null space or the nullity, which is the dimension of the null space?

4. Jun 13, 2012

### caliboy

I am looking for the nullspace not the nullity. The "nullspace" that I have in the fist post is the nullity. I have been stuyding this and am using the formula Axp=c and am not really understanding how I got a nullspace of (0,0)

I found A=
1x1+0x2=0
0x1+4x2=0
0x1+0x2=0

and a nullspace of:
0
0

still looking at B

5. Jun 13, 2012

### vela

Staff Emeritus
The null space is just the set of vectors that satisfy Ax=0. In your first example, the only solution is x=(0,0), so the null space is {(0,0)}, which is a vector space of dimension 0.

For your second problem, you found the nullity is 1, so the null space should turn out to be a vector space of dimension 1. That is, it should be the multiples of some vector. You want to figure out what that vector is by solving Bx=0.