# Homework 08 Solution

\$30.00

Category:

## 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. 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.

error: Content is protected !!