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rshalloo
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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