Finding rank(range) and nullspace of a matrix

[caliboy]

Homework Statement

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

Homework Equations

I used the Gauss elimination on both

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 completely wrong?

 

[Simon Bridge]

discussing how to find the null space.
https://www.physicsforums.com/showthread.php?t=100155
... how could you check to see if you are on the right track, without someone else telling you? (After all, we could be mistaken - how would you know?)

 

[vela]

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

 

[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 Ax_p=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

 

[vela]

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.