 
Summary: Problem 1 2 3 4 5 Bonus: Total:
Points 6 12 10 10 12 10 50+10
Scores
Mat 310 Linear Algebra Fall 2004
Name: Id. #: Lecture #:
Test 2 (November 05 / 60 minutes)
There are 5 problems worth 50 points total and a bonus problem worth up to 10 points.
Show all work. Always indicate carefully what you are doing in each step; otherwise it may not be possible
to give you appropriate partial credit.
1. [6 points] Let W1 and W2 be linear subspaces of a vector space V such that W1 + W2 = V and
W1 W2 = {0}. Prove that for each vector V there are unique vectors 1 W1 and 2 W2
such that = 1 + 2.
2. [12 points] Consider the vectors in R4
defined by
1 = (1, 0, 1, 2), 2 = (3, 4, 2, 5), 3 = (1, 4, 0, 9).
(a) [8 points] What is the dimension of the subspace W of R4
spanned by the three given
vectors? Find a basis for W and extend it to a basis B of R4
.
(b) [4 points] Use a basis B of R4
