What is the rank of a matrix with linearly dependent columns? And what's the null space?

Question

Let's say that a and b are independent vectors, while x and y are non-zero scalars. 

 

Suppose there is a matrix A = [a  xa  b  yb]. What would the rank be? Since there are 2 independent columns and 2 dependent columns, I think the rank should be 2?

 

 

Then for the second part, the title wording is a bit off. Basically, it's more like, what are 2 possible independent vectors which are in the null space of M (ie. Mx=0)? So far, I could think of only one, which is (x  -1  y  -1) transpose. I can't figure out another possible vector which isn't dependent on the first.

 

Thanks in advance!

Answer 1

The matrix has rank >= 2 by choice of a and b. On the other hand, any subset of 3 columns contains a linear dependence, so therefore the rank is less than 3.

 

Question 2 is testing rank(A) + nullity(A) = number of columns of A. So here, nullity(A) = 2, so we must have two linearly independent vectors in null(A), i.e. the null space has dimension 2. Think of this as the kernel of the linear transformation which is mapping a dimension 4 space to a dimension 2 space.

 

Your vector is indeed in the null space. Can you decompose it into two linearly independent vectors?

One can be (x -1 0 0)^T and another (0 0 y -1)^T