Linear Algebra

Math 235 - Summer 2015 Homework 1

Due Wednesday May 27 in class

Remember: In this course, you must always show reasoning for your answers. You can use any result we have proved in class, in textbook reading, or in a previous homework.

Problem 1

From Section 1.1 of the textbook, do Exercises 1 (parts a and b) and 3 (parts a and b).

Problem 2

Prove that, for all sets A, B, and C, if A ⊆ B then A∩C ⊆ B ∩C.

Hint: To get started, assume that A ⊆ B and let x be an arbitrary element of A∩C. Your goal is to then show that x ∈ B ∩C.

Problem 3

The union of the infinite family of sets A1,A2, . . . is defined by

∞⋃ i=1

Ai = {x : x ∈ Ai for some i ∈{1, 2, . . .}}.

Prove that

∞⋃ i=1

[i, i + 1] = [1,∞).

Hints: Show that the left-hand set equals the right-hand set by showing two things:

(i) x is in the left-hand set implies x is in right-hand set. (ii) x is in right-hand set implies x is in left-hand set.

First, choose an arbitrary element of the set on the left-hand side. By definition of the infinite union, it must belong to [i, i + 1] for some i. Explain why it must then belong to the set on the right-hand side. Second, choose an arbitrary element of the set on the right-hand side. Explain why it must belong to [i, i + 1] for some i. Then conclude that it must belong to the set on the left-hand side.

Problem 4

The intersection of the infinite family of sets A1,A2, . . . is defined by

∞⋂ i=1

Ai = {x : x ∈ Ai for all i ∈{1, 2, . . .}}.

Prove that

∞⋂ i=1

(−1/i, 1/i) = {0}.

Hints: We proceed as in problem 3 above. First, choose an arbitrary element of the set on the right-hand side. This element must be 0. Explain why it is in all of the sets (−1/i, 1/i) for all i. Second, choose an arbitrary real number which is not in the set on the right-hand side. Find some i for which this number is not in (−1/i, 1/i) (for example, 0.1 is not in the set (−1/11, 1/11), but you must handle the general case). Then conclude that this number is not in the set on the left-hand side.

Problem 5

Consider the function f : [−7, 10] → [−19, 15] given by f(x) = 2x− 5.

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(a) Prove that f is injective. (b) Prove that f is surjective. (c) Functions which are both injective and surjective are called bijective. Bijective functions have inverses.

Find a function f−1, called the inverse of f, such that (f−1 ◦ f)(x) = x for all x ∈ [−7, 10] and (f ◦f−1)(y) = y for all y ∈ [−19, 15]. Hints: You must first give an explicit formula for f−1(y), valid for all y ∈ [−19, 15]. Then you must show that (f−1 ◦f)(x) = x for all x ∈ [−7, 10] and (f ◦f−1)(y) = y for all y ∈ [−19, 15].

Problem 6

Find an example of a bijective (both injective and surjective) function f : (−1, 1) → R. You do not need to prove that f is bijective. Give an explicit formula for f(x), valid for all x ∈ (−1, 1). There are many possibilities. An appropriately scaled member of the trigonometric family will work.

Problem 7

Let F be a field. Prove that for all a,b,c ∈ F, the following statement is true: If a + b = c + b, then a = c.

(This is a cancellation law for fields given in Theorem C.1 on page 54.)

Problem 8 You are given a set V and two operations � and �. Explain whether (V,�,�) is a real vector space or not by providing a proof of the axioms or a counterexample to one of the axioms.

(a) V = Z, the set of integers, and � and � are the usual addition + and multiplication · in R, respectively. (b) V = [−1, 1] and for all u,v ∈ [−1, 1] and λ ∈ R,

u � v =

 

−1 , u + v ≤−1 u + v , −1 ≤ u + v ≤ 1

1 , u + v ≥ 1 and λ � u =

  −1 , λu ≤−1 λu , −1 ≤ λu ≤ 1

1 , λu ≥ 1.

(c) V = R2 and for all u = (u1,u2),v = (v1,v2) ∈ R2 and λ ∈ R:

u � v = (u1 + v1,u2 + v2) and λ � u = (λu1,u2).

Problem 9 You are given a subset S of a well known vector space V . Is S is a subspace of V ? If so, prove it. If not, provide a counterexample. The notation is identical to that of the book.

(a) V = F(R,R), the set of real-valued functions on R, and

S = {f ∈ V : lim t→∞

f(t) = 0}.

You may use knowledge about limits from Calculus. (b) V = F(R,R), the set of real-valued functions on R, and

S = {f ∈ V : f(t) = f(−t) for each t ∈ R}.

Notice S is the set of all real-valued even functions defined on the real line. (c) V is the set of sequences of real numbers and

S = {{an}∈ V : a1 + 3a2 ≤ 1}.

(d) V = C1(R), the set of continuously differentiable functions on R, and

S = {f ∈ V : f′(t) + (f(t))2 = 0 for all t ∈ R}.

You may use the fact that the set of continuously differentiable functions on R is a vector space over R with the usual definitions of function addition and scalar multiplication.

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Problem 10

(a) Exercise 1.3.10 from the textbook. In this problem, F is a field. (b) Prove that the set

S = {(x,y,z) ∈ R3 : x + y + z = b}

is a subspace of R3 if and only if b = 0. You may wish to use results you have proven in part (a) in your solution to this part.

Problem 11 Exercise 1.3.20 from the textbook. Use mathematical induction on the number of vectors n. This method is outlined below.

First, show that the claim in the problem is true for n = 1. That is, prove that for any vector w1 ∈ W and any scalar a1, a1w1 ∈ W . (You may use known properties of subspaces to do this.) This part of the proof is the base case.

Then, assume that the claim in the problem is true for some natural number n = k. That is, assume that for any vectors w1,w2, . . . ,wk ∈ W and any scalars a1,a2, . . . ,ak, a1w1 + a2w2 + · · · + akwk ∈ W . This is called the induction hypothesis, and is part of what is called the induction step.

Finally, prove that the claim is true for n = k + 1. That is, prove that for any vectors w1,w2, . . . ,wk+1 ∈ W and any scalars a1,a2, . . . ,ak+1, a1w1 + a2w2 + · · ·+ ak+1wk+1 ∈ W . This completes the induction step, and you’ll need to use your induction hypothesis and known properties of subspaces as you do it.

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