Question

Describe the null-space and range of the given linear transformation and say whether it is 1-1 or onto or both.
T: R^(nxn) mapped to R^(nxn) given as follows. For A in R(nxn), T(A) is obtained by replacing all diagonal entries in A by zeros and leaving all other entries unchanged.

* Thus far, I think the null o f T={[a], [a b/ cd]} and it is not one to one. I'm not sure how to find the range for this one though.

Answer 1

it is pretty clearly not one to one because matices with different entries along the diagonal but same entries everywhere else get sent to the same matrix

Answer 2

as for the null space, i would say it is the matrices with zeros outside the diagonal, but i am not sure what a nice way to characterize those is

Answer 3

actually i think they are called "diagonal matrices"

Answer 4

Perhaps just do a general matrix described with m columns and n rows and draw a diagonal \neq to 0, with all other elements =0.

Answer 5

should be square right?

Answer 6

Yes, I just looked it up, it would be a nxn diagonal matrix

Answer 7

i don't know what to say about the range, other than what it is, all matrices with zeros on the diagonal
is there a name for that?

Answer 8

Perhaps that could be described in terms of an inverse?

Answer 9

i don't really know
i would just say it in words

Answer 10

Non-invertible matrices?

Answer 11

Okay, the question does just say to describe, so I think it will be fine

Answer 12

yeah because there are non invertible matrices with diagonal entries, so that would not work

Answer 13

Oh, you're right. I will just describe it then. I do think that the range implies that this transformation is onto, because it equals T(A).