Vector Worksheets With Answers - Math 125, Exam 3
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19Name:
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Math 125 – Exam 3 – Version 1
November 16, 2005
60 total points possible
1. Let S be the set of all vectors of the form (5b + 2c, b, c) where b and c are arbitrary
numbers.
(a) (3pts) Find vectors u and v such that S = span u, v .
Solution:
v = (5b + 2c, b, c) = (5b, b, 0) + (2c, 0, c) = b(5, 1, 0) + c(2, 0, 1)
In other words, the vectors v are all linear combinations of the vectors (5, 1, 0) and (2, 0, 1).
Therefore, S = span (5, 1, 0), (2, 0, 1) .
3
(b) (2 pts) Why does this show that S is a subspace of
?
Solution: Recall that every vector space can be represented as the span of a set of basis
3
vectors. By definition, S is a vector space. It is a subspace of
since the vectors (5, 1, 0)
3
and (2, 0, 1) are vectors in
.
One could also explain how S satisfies the definition of a subspace as we did in the
solution of problem 2b of the 11 AM section of Quiz 5.
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