Properties Of Matrices with the same Column Space

[rshalloo]

Homework Statement

Suppose that A and B are 5 x 5 matrices with the same Column Space (image).
(a) Must they have the same columns?
(b) Must they have the same rank?
(c) Must they have kernels of the same dimension?
(d) Must they have the same kernel?
(e) If A is invertible, must B be invertible?
Justify each answer.

Homework Equations

Image/Column Space of A is the set of all vectors y of the form Ax=y for some vector x
Ax=y

The Attempt at a Solution

(a) No as when you multiply out the Ax for any A and x you get a set of linear combinations of x. ie first component of y = A11x1 +A12x2+...+A15x5
and 2 different linear combinations can be equal
(b) Yes? (don't know about this one its just a feeling)
(c) Yes because if the have the same Rank Then they must have same dimension of kernel and Rank+ dimension of Kernel = number of collumns
(d) Yes. well if the have the same collumn space for Ax=y for some x. But then Ax=0 for some x and so the have the same kernel
(e)Yes? (again just a feeling)

I'm just having a bit of trouble with this so any help would be much appreciated Cheers

 

[HallsofIvy]

Think of it this way- if you apply matrix A to (1, 0, 0, 0, 0) you get its first column as a vector. If you apply matrix A to (0, 1, 0, 0, 0) you get its second column as a vector, etc. In other words, the columns are precisely the result of applying A to every basis vector in turn. That is the same as saying that the "column space" is precisely the subspace of R⁵ that A maps all of R⁵ into- it is the "range" of A. If two matrices have the same "column space" they have the same range- they map R⁵ into the same subspace. Also remember the "range-kernel" relation: the dimension of the kernel plus the dimension of the range is equal to the dimension of the domain space- here, 5. If two matrices have the same column space, they have the same ranges and so the same dimensions- their kernels then must have the same dimension.

They do NOT necessarily have the same kernels. For example, the matrix
$$\begin{bmatrix}1 & 0 \\ 0 & 0\end{bmatrix}$$
and
$$\begin{bmatrix}0 & 1 \\ 0 & 0\end{bmatrix}$$
obviously have the same column space but the kernel of the first is spanned by <1, 0> and of the second by <0, 1>. They have the same dimension but are different subspaces of R².