Week 2 Homework

Problem Set 6 - More on Bases of Rn, Matrix Products

Learning Objectives:

• You should understand that n vectors form a basis of Rn ⇐⇒ rref [ ~v1 · · · ~vn

] = In ⇐⇒ every

vector ~x in Rn can be expressed as a linear combination of ~v1, . . . , ~vn in exactly one way.

• You should understand how matrix multiplication is related to composition of linear transformations.

• You should know when the product of two matrices is defined, be able to compute matrix products, and be familiar with the basic algebraic properties of matrix products.

The first problem below (#W1) is a warmup, which you need not turn in.

W1. Bretscher #2.3.14

1. Let A be the matrix of rotation in R2 counterclockwise by 30◦. Let B be the matrix of reflection in R2 over the x-axis.

(a) Find A.

(b) Find B.

Let T : R2 → R2 be defined by letting T (~x) be the vector obtained by first rotating ~x by 30◦ counter- clockwise and then reflecting the result over the x-axis.

(c) Without calculating anything, make a rough sketch of T (~e1) and T (~e2).

(d) Compute the appropriate matrix product to find the matrix of T .

(e) Use your answer to (d) to find T (~e1) and T (~e2).

4 Make sure your answers are consistent with your picture in (c)!

2. (a) Let Rθ be the matrix of rotation in R2 counterclockwise by an angle of θ, and let B be the matrix of reflection in R2 over the x-axis. (In the previous problem you computed BR30◦ .) Use matrix multiplication to verify that RθB = BR−θ.

(b) Let ~u be a unit vector, and let S~u be the matrix of reflection in R2 over the line determined by ~u. Let α be the angle between ~u and the positive x-axis. Give a geometric explanation of why S~u = RαBR−α. (You may assume that ~u lies in the first quadrant, so that 0 < α < 90

◦.)

(c) Let ~v be a second unit vector, and let β be the angle between ~v and the positive x-axis. Use parts (a) and (b) to show that S~vS~u = R2(β−α). Thus, the composition of two reflections in R2 is a rotation by twice the angle between the two lines.

3. (a) Bretscher #2.3.44

(b) Bretscher #2.3.46

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4. Neville is looking at 6 vectors ~v1, . . . , ~v6 in R6. He would like to express ~x =

 1 2 3 4 5 6

 as a linear combination of these vectors, and he thinks of two ways to do so:

~x = ~v1 + 4~v2 − ~v3 + ~v5 and ~x = −2~v1 − ~v2 + 7~v6.

Decide whether each of the following statements is definitely true or definitely false, or if there is not enough information to be sure; justify your answers.

(a) (~v1, . . . , ~v6) is a basis of R6.

(b) The reduced row-echelon form of the matrix [ ~v1 · · · ~v6

] is I6.

(c) ~v6 is in span(~v1, ~v2, ~v3, ~v4, ~v5).

(d) ~0 can be expressed as a linear combination c1~v1 + · · · + c6~v6 where the ci are not all 0 (it’s fine for some of the ci to be 0, just not all of them).

Note: Of course, ~0 can be expressed as the linear combination 0~v1 + 0~v2 + 0~v3 + 0~v4 + 0~v5 + 0~v6, but this is not so interesting, so it’s called the trivial linear combination of ~v1, . . . , ~v6. We’re interested in seeing whether ~0 can be expressed as a nontrivial linear combination of ~v1, . . . , ~v6, i.e., as any linear combination of ~v1, . . . , ~v6 other than 0~v1 + · · ·+ 0~v6.

(e) There are infinitely many ways to express ~x as a linear combination of ~v1, . . . , ~v6.

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