 
Summary: Math 310: Midterm 1
October 10, 2006
Name: ID number:
There are 5 questions worth a total of 100 points, plus one small bonus question
worth 10 points. Please justify all your statements, and write neatly so that we can
read and follow your answers. Continue your answers on the back of the pages. Also,
please turn off cell phones.
Question 1. (20 points) (i) Define a subspace of a vector space.
(ii) Suppose that V is a finite dimensional vector space and that W V is a subspace
such that dim W = dim V . Prove carefully that W = V .
1 20pt
2 20pt
3 20pt
4 30pt
5 10pt
Total 100pt
bonus 10pt
Question 2. (20 points) (i) Let (v1, . . . , vn) be a list of vectors in V . What does it
mean to say that this list is linearly independent? Give the formal definition.
(ii) Give an example of a list of three vectors in R4
