Homework 08 Solution

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

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

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.

4. (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. 2. 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 : Rⁿ ! R^m is a linear transformation.

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

6. 2. 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

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

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

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

bination of ~e₁; ~e₂; : : : ; ~e_n?

7. 2. Suppose that ~x can be written as a linear combination of ~e₁; ~e₂; : : : ; ~e_n in two different ways. That is,

~x = c₁~e₁ + c₂~e₂ + + c_n~e_n; and ~x = d₁~e₁ + d₂~e₂ + + d_n~e_n

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; : : : ; ~e_ng to be a basis has been violated.

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