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Lecture 3

MTH 341 Lecture Notes - Lecture 3: Row Echelon Form, Elementary Matrix, Solution SetExam


Department
Mathematics
Course Code
MTH 341
Professor
Rockwell, Dan
Lecture
3

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Math 341 - Monday, April 8th, 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.
(1) True or False: If two matrices Aand Bare equivalent then they have identical reduced
row-echelon forms.
(2) Are the following two matrices, Aand Bequivalent? If they are equivalent, show via
elementary row operations that they are equivalent.
A=2
4
23
65
25
3
5
B=2
4
23
04
00
3
5
(3) Do you think the two matrices in problem 2 are equal?
(Note: Equal and Equivalent have very dierent meanings for matrices. We will look
at matrix equality in our near future, Definition 2.4 page 54)
(4) Why do homogenous systems always have at least one solution? (What is a geometric
way you could think about this? What are you visualizing geometrically?)
(5) What is the name of this guaranteed solution? (Does that name make sense to your
group?)
(6) If a system of homogeneous linear equations has mequations and nunknowns when
will the system be guaranteed to have infinitely many solutions (i.e. non-trivial solu-
tions exist).
(7) Find the nontrivial solutions for the following homogeneous system of equations.
x+y+z=0
2xy+z=0

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Math 341 - Monday, April 8th, 2019 Page 2 of 3
(8) What are your basic solution(s) for problem 7?
(9) Find the nontrivial solutions for the following homogeneous system of equations.
x+y+z=0
2x+2y+2z=0
The following questions may assist you as you try to accomplish the given task:
Are there more unknowns than equations? (Since this is a homogeneous system what
does this answer to the question above tell us?)
How many parameters do you need to describe the solution set?
(10) What are your basic solution(s) for your solution set to the system in problem 9?
The definition from our book of a linear combination of column vectors is found on
page 30. Below is an equivalent definition that we can also use.
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.
(11) Show that the vector ~u=2
4
4
2
1
3
5is a linear combination of the three vectors
~e1=2
4
1
0
0
3
5
~e2=2
4
0
1
0
3
5
~e3=2
4
0
0
1
3
5.(~e1,~e
2,and ~e3are the standard unit vec-
tors.)
Another way to phrase this is find scalars c1,c
2,and c3such that ~u=c1
~e1+c2
~e2+c3
~e3.
(12) Show that the vector ~u=2
4
4
2
1
3
5is a linear combination of the three vectors
~v1=2
4
1
0
1
3
5
~v2=2
4
0
1
0
3
5
~v3=2
4
0
0
2
3
5.
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