Math 317: Linear Algebra
Worksheet 1: 1.1,1.2,1.4
Fall 2015
Name:
1. (a) Suppose that x, y ∈ Rn are nonparallel vectors. Prove that if sx + ty = 0 then
s = t = 0. (Hint: It may be useful to use contradiction here. That is, assume
that either s 6= 0 or t 6= 0, and show that this is impossible by contradicting
the original hypothesis.)
(b) Use (a) to prove that if ax + by = cx + dy, then a = c and b = d.
2. Using only those properties in Section 1.1, Problem 28, prove that if a + c = b + c
for some vectors a, b, c ∈ Rn , then a = b. This is the cancellation property that
some of you used (without proof) in your homework assignment.
3. Let u, v ∈ Rn and suppose that u − v is orthogonal to u + v. Prove that u and v
have the same magnitude.
4. (a) For some x ∈ Rn , show that x · y = x · z does not necessarily imply that
y = z by coming up with a suitable example. (Suggestion: Look for examples
in R2 . Is it possible for a vector x to be orthogonal to two different vectors? )
(b) Suppose that x · y = x · z for all x ∈ Rn . Show that y = z.
5. Determine if (3, 4, −1, 6) lies in span {(1, 2, −1, 2), (−2, 3, 1, −1), (−1, 3, 2, 1)}.
6. Determine the reduced row echelon form of A and give the solution to Ax = 0 in
standard form.


2 −2 4
A = −1 1 −2
3 −3 6
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