# Geometric description of the nullspace

1. Nov 8, 2009

### Tomblue

1. The problem statement, all variables and given/known data
general form of solutions to Ax=b
Consider matrix A=
{[ 2 -10 6 ]
[ 4 -20 12 ]
[ 1 -5 3 ]}
Find a basis for the nullspace of A. Give a geometric description of the nullspace of A.

3. The attempt at a solution
I found the basis for the nullspace of A to be
{[-3 5]
[0 1 ]
[1 0 ]}
The thing i dont understand is how to give a geometric description of the nullspace of A. If someone could help to explain how i would start to go about doing this that would be awesome because i'm not quite sure i understand the question.

2. Nov 8, 2009

### Quantumpencil

Are you saying that your null space is spanned by {-3, 0, 1} and {5, 1, 0}? Your notation is a bit confusion to me; but I believe that they are asking you to describe what kind of "Space" this is. What do two vectors span?

3. Nov 8, 2009

### Tomblue

I was trying to say that the vectors {-3,0,1} and {5,1,0} form the basis for the nullspace of A and that i'm not seeing how to give a geometric description of the nullspace of A.

4. Nov 8, 2009

### Staff: Mentor

How many linearly independent vectors does it take to span a line? A plane? A three-dimensional space?

5. Nov 8, 2009

### Tomblue

one linearly independent vector to span a line, two linearly independent vectors to span a plane, and 3 linearly independent vectors to span a 3-dimensional space, and so forth any n vectors that span an n-dimensional space are going to be linearly independent. So i see i'm going to have 2 linearly independent vectors and therefore the dim(W) is going to be 2 dimensional like i thought. Question: Can the vectors ever be identical say that v1=v2

6. Nov 8, 2009

### Staff: Mentor

Linearly independent vectors can't be identical or even multiples of one another.