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MATH 141: Linear Analysis I Homework 08 solved

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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 → 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.