Linear Algebra HW
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chapter_4.pdf
Hussam Malibari Heckman MAT 242 Spring 2017 Assignment Chapter 4 due 04/04/2017 at 11:59pm MST
In some of the problems in this chapter, you will be asked to enter a basis for a subspace. You should do this by placing the entries of each vector inside of brackets, and giving a list of these vectors, separated by commas. For instance, if your basis is
12
3
, 11
1
,
then you would enter [1,2,3],[1,1,1] into the answer blank.
1. (1 point) Which of these vectors can be written as a linear
combination of
−7 1 −1 2
and
7 −2 7 −6
?
• A.
35 −7 17 −18
• B.
119 −19 62 −72
• C.
18 −36 46 −36
• D.
−7 −7 47 −30
2. (1 point) Which of the following sets of vectors are lin- early independent?
• A. {[
6 9
] ,
[ −6 −9
]} • B.
−58
0
, 2−7
0
, 4−9
0
• C. {[
−2 5
] ,
[ 3 −8
] ,
[ 6 9
]} • D.
{[ −7 4
]} • E.
1−8
4
, 09
−6
, 52
3
• F. {[
0 0
] ,
[ −3 8
]}
3. (1 point) Let A =
20
−2
, B =
21
−3
, and C =
−40
7
.
? 1. Determine whether or not the three vectors listed above are linearly independent or linearly dependent.
2. If they are linearly dependent, find a non-trivial linear combi- nation of A,B,C that adds up to~0. Otherwise, if the vectors are linearly independent, enter 0’s for the coefficients.
A+ B+ C = 0.
4. (1 point) Let A =
−3 12 −7 6
, B =
0 2 −1 −1
, C =
−1 4 −1 2
,
and D =
−2 10 −5 3
.
? 1. Determine whether or not the four vectors listed above are linearly independent or linearly dependent.
2. If they are linearly dependent, find a non-trivial linear com- bination which adds up to the zero vector. Otherwise, if the vectors are linearly independent, enter 0’s for the coefficients.
A+ B+ C+ D = 0.
1
5. (1 point) Find a basis for the subspace of R4 spanned by the following vectors.
−1 0 0 0
,
−2 −1 1 −2
,
−5 −2 2 −4
,
−1 2 1 −1
Answer:
6. (1 point) Find a basis for the subspace of R4 consisiting of all vectors of the form
x1
9x1 + x2 8x1 − 8x2 −4x1 − 3x2
Answer:
7. (1 point) Find a basis for the subspace of R3 consisting of
all vectors
x1x2
x3
such that −6x1 − 7x2 − 6x3 = 0.
Hint: Notice that this single equation counts as a system of linear equations; find and describe the solutions.
Answer:
8. (1 point) Consider the ordered basis B of R2 consisting of
the vectors [
3 1
] and
[ −2 3
] (in that order). Find the vector~x
in R2 whose coordinates with respect to the basis B are [
−4 1
] .
~x = [ ]
9. (1 point) The set B = {[
−5 −2
] ,
[ 15 9
]} is a basis for
R2. Find the coordinates of the vector ~x = [
30 15
] with respect
to the basis B:
[~x]B = [ ]
10. (1 point) Suppose that A is a 7 × 5 matrix.
(a) A vector in the null space of A has entries in it. (b) A vector in the row space of A has entries in it. (c) A vector in the column space of A has entries in it.
11. (1 point) Let A =
1 0 0 −4 −2 4 0 0 −16 −8 0 1 0 1 −3 0 1 −2 −9 3
. Find
a basis for the row space of A, a basis for the column space of A, a basis for the null space of A, the rank of A, and the nullity of A. (Note that the reduced row echelon form of A is
1 0 0 −4 −2 0 1 0 1 −3 0 0 1 5 −3 0 0 0 0 0
.)
Row Space basis: Column Space basis: Null Space basis: Rank: Nullity:
12. (1 point) Let A =
1 1 −1 0 0 0 4 4 −4 0 0 0 0 0 0 0 −2 −3 0 0 0 0 1 3 0 0 0 −1 3 −3
.
Find a basis for the row space of A, a basis for the column space of A, a basis for the null space of A, the rank of A, and the nullity of A. (Note that the reduced row echelon form of A is
1 1 −1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0
.)
Row Space basis: Column Space basis: Null Space basis: Rank: Nullity:
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