I'm pretty sure that D and F are linear, but i reallly don't know for sure how to prove it on paper. I'm not looking for answers here, just help on how to solve them.
(link to the problems)
If anyone could explain how to do this, that would be amazing. I know that to be linear, a transform has to satisfy T(u + v) = T(u) + T(v) and T(ku) = kT(u), but i'm not sure how to apply it in all of these situations.
Thanks in advance!
A) What is ? What is ?
B) I am not sure what the represents. If it is supposed to be , then
to show the required properties consider:
i) if is a given matrix and , then .We want to show this equals . There is a certain property we can use to remove the parentheses...
ii) It has probably already been shown that scalars commute with matrices...
C) postponed while we consider
D) Note that the logic for B (or my interpretation of it) still holds if .
Now for C), it should be easy to see that the composition of linear maps is linear (just apply the definitions!) so note that if , , and then are linear maps by the previous parts, and thus it is linear!
E) This is easy if you think about what the transpose means. We simply reflect across the diagonal. But then since adding and scaling are entrywise (the entry of is the entry of plus the entry of ). So transpose the sum, sum the transposes, see what happens. Similarly since scaling affects all entries the same, transposing does not affect scaling...
F) For this problem, all that remains to be shown is that the difference of linear maps is linear. Simply write everything out and see what happens