Math 317: Linear Algebra Homework 1 Due: January 22, 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: 1,2,5,6,7,8,20,25,28 Section 1.2: 1,2,6,11,13,17 Turn in the following problems. 1. Section 1.1, Problem 13 2. Section 1.1, Problem 21 3. Section 1.1, Problem 22 4. Section 1.1, Problem 29 5. 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) 6. Section 1.2, Problem 7 7. Section 1.2, Problem 18 1