Math 317: Linear Algebra
Homework 1
Due: September 4th, 2015
The following problems are for additional practice and are not to be turned in: (All
problems come from Linear Algebra: A Geometric Approach, 2nd Edition by ShifrinAdams.)
Exercises: Section 1.1: 1,2,5,6,7,12,21,22,28
Section 1.2: 1,2,6,7,11,13,17
Turn in the following problems.
1. Section 1.1, Problem 13
2. Section 1.1, Problem 20
3. Section 1.1, Problem 29
4. In class, we derived the definition of the dot product between two vectors, say a
and b both from a geometric point a view, and an algebraic point of view. In this
exercise, you will derive the algebraic definition of a dot product using the law of
cosines. Starting with the vectors a = (a1 , a2 ) and b = (b1 , b2 ), draw a triangle (in
Quadrant I) whose sides are the magnitude of a and b. Write the third side of the
triangle in terms of a and b. Let φ be the angle opposite of the side of the triangle
that you have written in terms of a and b. The law of cosines asserts that
c2 = kak2 + kbk2 − 2kakkbk cos φ,
(1)
where c is the third side of the triangle that you have written in terms of a and b.
Using (1) and the geometric form of the definition of dot product, show that
a · b = a1 b 1 + a2 b 2
(2)
5. Section 1.2, Problem 18
6. The Cauchy-Schwarz Inequality |u · v| ≤ kukkvk is equivalent to the inequality
we get by squaring both sides: (u · v)2 ≤ kuk2 kvk2 . In R2 , with u = (u1 , u2 ) and
v = (v1 , v2 ), this becomes
(u1 v1 + u2 v2 )2 ≤ (u21 + u22 )(v12 + v22 ).
Prove this algebraically.
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