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 .

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