MTH 341 Study Guide - Final Guide: Augmented Matrix, Linear Independence, Linear Combination
15 May 2019
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Department
Course
Professor
Math 341 - Wednesday, April 24th, 2019 Page 1 of 3
Name:
Directions: Discuss and solve the following problems in your groups. You are encouraged
to write out your work on additional pieces of paper.
Definition:Asetofnon-zerovectors{~u1,~u
2,...,~u
k}in Rnis said to be linearly indepen-
dent if the only solution to the following homogeneous system of equations is the trivial
solution. (i.e. all of the aicoefficients are zero)
k
X
i=1
ai
~ui=~
0
(1) Consider the following system of linear equations.
8
<
:
x−y+z=0
2x+3y+z=0
2x−2y+2z=0
This system of equations can be represented in a variety of ways. Below are a couple
of equivalent representations of the system above.
(a) (Augmented Matrix Notation) 2
4
1−110
2310
2−220
3
5
(b) (Matrix-Vector Notation) A~x=
~
bwhere
A=2
4
1−11
231
2−22
3
5,~x=2
4
x
y
z
3
5,and
~
b=2
4
0
0
0
3
5
(c) (Vector Notation) x2
4
1
2
2
3
5+y2
4
−1
3
−2
3
5+z2
4
1
1
2
3
5=2
4
0
0
0
3
5
The representation in 1c is the most valuable when thinking about linear indepen-
dence/dependence.
Question: Are the vectors 2
4
1
2
2
3
5,2
4
−1
3
−2
3
5,and2
4
1
1
2
3
5linearly dependent or lin-
early independent?
If the vectors are linearly dependent find a non-trivial linear combination of the
vectors that is equal to the zero vector.
LECTURE NOTES
1)
fly ),,fly)linearly dependent or independent ?
AX=o
Ilike .t.si :tyii¥H¥÷¥¥
OOOO/Rt Rt →Rl OOOO
-2 -3 -I0-2 2-2 OOl¥0
z
O-5IOOOOOlO450
since Ato ,one solution is
'skit 's "Ito'
Pivot columns -1 linearly independent
2) What is the span off 1/3/11 !)}?Two linearly independent vectors have a
span of aplane .
if you dropped the last RREF column ,you
Aplane in 3D going through the origin .get linearly independent vectors (two )
which span aplane in the 3-D .
3) it -
-(ll05
'N' 41 005'F' fo Ol'T
ORl HR2
l!!!!).im/o'78/8/Yoewmnasreatienepa.:oYoionwaemne.naen-
became an
OOlO
-IOOO
1€
OlOO
Math 341 - Wednesday, April 24th, 2019 Page 2 of 3
Definition:Let~v1,...,~v
kand ~ube column vectors.
Then ~uis said to be a linear combination of the vectors {~v1,...,~v
k}if there exist
scalars c1,...,c
ksuch that
~u=
k
X
i=1
ci
~vi.
Definition: The span of a set of vectors {~v1,...,~v
k}is the set of all linear combi-
nations of the vectors {~v1,...,~v
k}.Thenotationwewilluseforthespanofasetof
vectors is
span (~v1,...,~v
k)
(2) What is the span of 8
<
:
2
4
1
2
2
3
5,2
4
−1
3
−2
3
5,2
4
1
1
2
3
59
=
;
?
(These are the vectors from problem 1)
Describe this however you wish. If you are not sure how to describe this span ask for
advice.
(3) Show that the following set of vectors is either linearly independent or linearly de-
pendent. If the vectors are linearly dependent find a non-trivial linear combination
that equals the zero vector.
~u=[1,1,0]T~w=[1,0,0]T~p=[0,0,1]T
(4) Show that the following set of vectors is either linearly independent or linearly de-
pendent. If the vectors are linearly dependent find a non-trivial linear combination
that equals the zero vector.
~u=[−2,3,2]T~w=[2,1,0]T~p=[−2,5,3]T
(5) What is the span of {~u=[−2,3,2]T,~w=[2,1,0]T,~p=[−2,5,3]T}?Describethis
however you wish. If you are not sure how to describe this span ask Dan for advice.
(6) Suppose {~v1,~v
2}is a linearly independent set in Rn.Is{~v1,~v
1+~v2}also linearly
independent?
(7) Let {~v1,~v
2,~v
3,~v
4}be a subset of Rn.Forwhatvaluesofn≥1canyouguarantee
that the set of vectors is linearly dependent? Justify your answer.
2linearly independent vectors es #of pivot columns .
