Homework 2
Problems you need to submit to gradescope:
1. Let S and S ′ be two list of vectors.
(a) Show that if each element of S ′ is in the span(S), then span(S ′ ) is a
subset of span(S).
(b) Use (a) to show span(S ′ ) = span(S) if and only if each vector in S ′
is in span(S) and each vector in S is in span(S ′ ).
2. Textbook 2A: 13
3. Textbook 2B: 5
4. Textbook 2B: 10
5. Let V1 , ...Vk be subspaces of V , show that if V1 + V2 + ... + Vk is a direct
sum then dim(V ) = dim(V1 )+dim(V2 )+...+dim(Vk ). (Hint: think about
how to generalize the result of 4 to direct sum of more subspaces)
6. Textbook 2C: 8
7. Textbook 2C: 18
Note: You need to write out and explain your work to receive full
credit, also don’t round up your answer. (For example don’t simplify
trig or inverse trig into decimal).
Extra problems to try yourself:
Textbook 2A: 4, 8, 10, 11, 14, 17, 18
Textbook 2B: 1, 3, 6, 8, 9
Textbook 2C: 2, 3, 6, 9, 11, 14, 16
Even though the extra problems are not mandatory, I highly recommend you to at least look at the problem (especially the red ones)
to see if you know how to do it, since the quiz and/or exam problems
might come from those problems.
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