Using Dimension Formula of a Matrix

[tatianaiistb]

Homework Statement

If VΩW contains only the zero vector, then dim(V+W)+dim(VΩW) = dim(v) + dim(W) becomes dim(V+W) = dim(v) + dim(W). Check this when V is the row space of A, W is the nullspace of A, and the matrix A is mxn of rank r. What are the dimensions?

Homework Equations

dim(V+W)+dim(VΩW) = dim(v) + dim(W)

The Attempt at a Solution

I don't even know how to start this problem.

I do know that dim(V+W) + dim (VΩW) = rank of [A B] + nullity of [A B]

Can anyone help? Thanks!

 

[HallsofIvy]

It would help if you told us what thes symbols mean. I presume that V and W are vectors but what is \Omega and what do you mean by V\Omega W?

 

[tatianaiistb]

I'm sorry! That symbol means V intersects W... And yes, V and W are vectors.

 

[Mark44]

I'm sorry! That symbol means V intersects W... And yes, V and W are vectors.

No, V and W are vector spaces.