Description

Construct examples of linear transformation that satisfy the following requirements. If no such examples are possible, explain why. (Hint: Problems #68 of Homework 06 help you connect onetoone or onto linear transformations to properties of matrices.)

onetoone but not onto
onto but not onetoone
both onetoone 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 R^{3} are actually subspaces? For each subspace you find, find a basis for that subspace. Describe your reasoning.

~
^{2}b_{1}
3
(a)
^{b}2
5
with first component b
= 0.
The plane of vectors b =
(b)
~
^{4}b_{3}
1
The plane of vectors b with first component b_{1} = 1.
(c)
~
= 0
(notice that this is the union of two subspaces, the plane b_{2} = 0 and the
The vectors b with b_{2}b_{3}
plane b_{3} = 0).
^{3} and
2
3_{.}
1
2
(d)
All linear combinations of two given vectors ^{2}1
0
(e)
~
4_{0}
5
4
1
5
The plane of vectors b that satisfy b_{3} b_{2} + 3b_{1} = 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


Any straight line in R^{2} is a vector subspace of R^{2}.



Any twodimensional plane going through the origin in R^{3} is a vector subspace of R^{3}.


(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.

(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 = 4_{6}
2
0
2
3
1
7
1


If A is an mbyn matrix with rank r. What is dim N(A)? What is dim C(A). Explain your reasoning. (Hint: review the worksheet.)


(postponed to next week) T : R^{n} ! R^{m} is a linear transformation.


Is ker(T ) a subspace of R^{n}?. Explain your reasoning. If yes, how can you find a basis for ker(T )?



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

Follow the steps below to prove the theorem: If f~e_{1}; ~e_{2}; : : : ; ~e_{n}g is a basis for R^{n}, then any vector ~x in R^{n}
can be written as a linear combination of ~e_{1}; ~e_{2}; : : : ; ~e_{n} in a unique way.


Which requirement for f~e_{1}; ~e_{2}; : : : ; ~e_{n}g to be a basis ensures that ~x can be written as some linear com

bination of ~e_{1}; ~e_{2}; : : : ; ~e_{n}?


Suppose that ~x can be written as a linear combination of ~e_{1}; ~e_{2}; : : : ; ~e_{n} in two different ways. That is,

~x = c_{1}~e_{1} + c_{2}~e_{2} + + c_{n}~e_{n}; and ~x = d_{1}~e_{1} + d_{2}~e_{2} + + 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~e_{1}; ~e_{2}; : : : ; ~e_{n}g to be a basis has been violated.
(c) Explain briefly why putting parts (a) and (b) together leads to a proof of the theorem.