Homework 08 Solution

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  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

3

! R

R

2

3

! R

R

3

2

! R

2. (Strang x2.1 #2) Which of the following subsets of R3 are actually subspaces? For each subspace you find, find a basis for that subspace. Describe your reasoning.

~

2b1

3

(a)

b2

5

with first component b

= 0.

The plane of vectors b =

(b)

~

4b3

1

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

(c)

~

= 0

(notice that this is the union of two subspaces, the plane b2 = 0 and the

The vectors b with b2b3

plane b3 = 0).

3 and

2

3.

1

2

(d)

All linear combinations of two given vectors 21

0

(e)

~

40

5

4

1

5

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) f~g is a vector subspace of any n, where ~ has zeroes as coordinates.0R0n

    1. Any straight line in R2 is a vector subspace of R2.

    1. Any two-dimensional plane going through the origin in R3 is a vector subspace of R3.

  1. (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.

  1. (revised on Wednesday) The dimension of a vector subspace W, denote by dim W, is defined to be the number of vectors in its basis.

2

3

3

1

0

1

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

(a) For the matrix in the worksheet, A = 46

2

0

2

3

1

7

1

    1. 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.)

  1. (postponed to next week) T : Rn ! Rm is a linear transformation.

    1. Is ker(T ) a subspace of Rn?. Explain your reasoning. If yes, how can you find a basis for ker(T )?

    1. Is range(T ) a subspace of Rm?. 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

  1. Follow the steps below to prove the theorem: If f~e1; ~e2; : : : ; ~eng is a basis for Rn, then any vector ~x in Rn

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

    1. Which requirement for f~e1; ~e2; : : : ; ~eng to be a basis ensures that ~x can be written as some linear com-

bination of ~e1; ~e2; : : : ; ~en?

    1. 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 f~e1; ~e2; : : : ; ~eng to be a basis has been violated.

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


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