Problem 1
Give an example of a linear map with and .
Consider
Then the null space is all vectors of the form , giving . The range is all vectors of the form , which clearly gives . Note that this also fulfills the fundamental theorem of linear maps, since .
Problem 2
Suppose are such that . Prove that .
We have:
As , we have
for any in .
This means that for all . The final operation would be , and since linear maps take 0 to 0, we have .
Problem 3
Suppose is a list of vectors in . Define by
- (a) What property of corresponds to spanning ?
- (b) What property of corresponds to being linearly indepedent?
(a)
Problem 4
Show that is not a subspace of .
Problem 5
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