CS 680 PSet 1: Linear Algebra Self-Assessment

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Question 1:

(a) Given points p1 = (1, 6, 5) and p2 = (5, 3, −7), solve for v2 the vector from p1 to p2.

(b) Given a third point p3 = (1, 6, 4), solve for v3 the vector from p1 to p3.

(c) Find the values for the magnitudes of v2 and v3.

(d) Solve for the unit vectors in the directions of v2 and v3.

Question 2:

(a) Solve for the vector (cross) product v2 × v3.

(b) Solve for v3 × v2.

(c) Solve for the scalar (dot) product v3 · v2.

Question 3:

(a) If two vectors u, v ∈ <n are orthogonal, what is the value of their scalar (dot) product?

(b) If two vectors u, v ∈ <n are parallel, what is the value of their cross product?

Question 4:

Which of the following are unit vectors? (a) ( 1

2

, −

1

2

, 0) (b) (0, −1, 0) (c) 1

7

(−2, 3, 6)

Question 5:

We are given two non-zero vectors u, v ∈ <3

. Assume the angle between u and v satisfies 0 < θ < π

2

.

Use dot products and/or cross products of u and v to give expressions for:

(a) cos θ (b) sin θ (c) A vector perpendicular to both u and v.

Question 6:

Given three square matrices Q, R, S ∈ <n×n

, which statements are true in general? If the

statement is false, please correct it.

(a) (QRS)

−1 = Q−1R−1S

−1

1

(b) QR = RQ

(c) (QRS)

T = S

T RT QT

(d) (R + S)Q = SQ + RQ

Question 7:

Given a square matrix A ∈ <n×n whose columns form an orthonormal basis:

(a) What is the dot product of any pair of columns in A?

(b) What is the inverse of A?

2guu