## Description

1. Construct examples of linear transformation that satisfy the following requirements. If no such examples

are possible, explain why. (Hint: Problems #6-8 of Homework 06 help you connect one-to-one or onto

linear transformations to properties of matrices.)

one-to-one but not onto onto but not one-to-one both one-to-one and onto

R

2 → R

2

R

3 → R

3

R

2 → R

3

R

3 → R

2

2. (Strang §2.1 #2) Which of the following subsets of R

3 are actually subspaces? For each subspace you find,

find a basis for that subspace. Describe your reasoning.

(a) The plane of vectors ~b =

b1

b2

b3

with first component b1 = 0.

(b) The plane of vectors ~b with first component b1 = 1.

(c) The vectors ~b with b2b3 = 0 (notice that this is the union of two subspaces, the plane b2 = 0 and the

plane b3 = 0).

(d) All linear combinations of two given vectors

1

1

0

and

2

0

1

.

(e) The plane of vectors ~b that satisfy b3 − b2 + 3b1 = 0.

3. Determine each of the following statements true or false. Explain your reasoning.

(a) {~0} is a vector subspace of any R

n, where ~0 has n zeroes as coordinates.

(b) Any straight line in R

2

is a vector subspace of R

2

.

(c) Any two-dimensional plane going through the origin in R

3

is a vector subspace of R

3

.

4. (added on Wednesday) Finish the worksheet in lecture titled ”Basis for N(A) and C(A)”, a copy of which is

posted on CatCourses. Turn in a digital copy of your solution together with the rest of this homework set,

and bring a hard copy of your solutions to class on Monday.

5. (revised on Wednesday) The dimension of a vector subspace W, denote by dimW, is defined to be the number

of vectors in its basis.

(a) For the matrix in the worksheet, A =

3 1 0 −1

3 1 −7 1

6 2 0 −2

. what is dim N(A)? What is dim C(A)?

(b) If A is an m-by-n matrix with rank r. What is dim N(A)? What is dim C(A). Explain your reasoning.

(Hint: review the worksheet.)

6. (postponed to next week) T : R

n → R

m is a linear transformation.

(a) Is ker(T) a subspace of R

n?. Explain your reasoning. If yes, how can you find a basis for ker(T)?

(b) Is range(T) a subspace of R

m?. Explain your reasoning. If yes, how can you find a basis for range(T)?

(Hint: Connect ker(T) and range(T) to column space and nullspace of some matrix.)

MATH 141: Linear Analysis I Homework 08 Fall 2019

7. Follow the steps below to prove the theorem: If {~e1, ~e2, . . . , ~en} is a basis for R

n, then any vector ~x in R

n

can be written as a linear combination of ~e1, ~e2, . . . , ~en in a unique way.

(a) Which requirement for {~e1, ~e2, . . . , ~en} to be a basis ensures that ~x can be written as some linear combination of ~e1, ~e2, . . . , ~en?

(b) Suppose that ~x can be written as a linear combination of ~e1, ~e2, . . . , ~en in two different ways. That is,

~x = c1~e1 + c2~e2 + · · · + cn~en, and ~x = d1~e1 + d2~e2 + · · · + dn~en

where all the c’s are not the same as all the d’s. By calculating ~x − ~x, show that one requirement for

{~e1, ~e2, . . . , ~en} to be a basis has been violated.

(c) Explain briefly why putting parts (a) and (b) together leads to a proof of the theorem.