Row and Column space questions.

[seang]

Hey, I was looking for help on these questions dealing with row and column spaces...

1. Prove that the linear system Ax = b is consistent IFF the rank of (A|b) equals the rank of A.

2. Show that if A and B are nxn matrices, and N(A-B) = R^n, then A = B

The first one I can't get much of a handle on. I can sort of feel like its going to have to do something with b lying in the column space of A? maybe? I can't quite get there. The second one I think I understand: N(0) corresponds to 0x = x, which is any vector in R^n, or something along those lines.

And also, I have a general question: If a matrix has linearly independent column vectors, under what conditions are its row vectors linearly independent?

Thanks for any help

 

[StatusX]

You're on the right track with the first one. The rank of a matrix is the dimension of the column space, or in other words, the number of columns that are linearly independent. So if the rank doesn't change when you add another column vector, what is true of this new column with respect to the old ones?

For the second, you want to show that the only matrix that has all of R^n as its null space is the 0 matrix. That is, show that if there is a single non-zero element in A, then there is some vector x with Ax\neq0.

There is a theorem that the dimension of the column space of a matrix is the same as the dimension of the row space. So the answer to your question is that it depends on the size of the matrix (ie, if it's nxm, is n>m, n=m, or n<m?)