Mathl Ill - Mathematics 1|
cfw_I,_].-',3]= [5 +t,-'-lEt-.t], t ED _
To discuss the geometric interpretation of this solution let
1 . .3. ._3.
l=-'-l,= IE=EEl-t'll:il-'I=E
1'.
'l: I I
:3: 4 2
Then we can sa1,r that b can be written as a linear combinat
MA 405 Introduction to Linear Algebra and Matrices
Quiz 7 Solution
March 30, 2010
Instructions: Show all work relevant to the solution of each problem. i.e. no credit will be given for just
the answers. There is one problem which carries a total 10 points
MA 405 Introduction to Linear Algebra and Matrices
Quiz 8 Solution
April 11, 2010
Instructions: Show all work relevant to the solution of each problem. i.e. no credit will be given for just
the answers. There is one problem which carries a total 10 points
MA 405 Introduction to Linear Algebra and Matrices
Quiz 6 Solution
March 7, 2010
Instructions: Show all work relevant to the solution of each problem. i.e. no credit will be given for just
the answers. There are two problems which carry a total 10 points.
MA 405 Introduction to Linear Algebra and Matrices
Quiz 5
February 26, 2010
Instructions: Show all work relevant to the solution of each problem. i.e. no credit will be given for just
the answers. There are two problems which carry a total 10 points. Good
MA 405 Introduction to Linear Algebra and Matrices
Quiz 4
February 19, 2010
Instructions: Show all work relevant to the solution of each problem. i.e. no credit will be given for just
the answers. There are three problems which carry a total 10 points. Go
MA 405 Introduction to Linear Algebra and Matrices
Last Quiz! (9) Solution
April 16, 2010
Instructions: Show all work relevant to the solution of each problem. i.e. no credit will be given for just
the answers. There are two problems which carry a total 1
MA 405-002 Spring 2014
Jacob Norton
Quiz #1 - January 10, 2014
Answer the questions in the space provided on the page. Show all your work; the correct answer without any
work receives no credit. Keep your work neat!
SOLUTION KEY
1. (10 points) Solve the f
MA 405-002 Spring 2014
Jacob Norton
Quiz #4 - February 7, 2014
Answer the questions in the space provided on the page. Show all your work; the correct answer without any
work receives no credit. Keep your work neat!
PRINT YOUR NAME:
1. (10 points) Conside
MA 405-002 Spring 2014
Jacob Norton
Quiz #5 - February 19, 2014
Answer the questions in the space provided on the page. Show all your work; the correct answer without any
work receives no credit. Keep your work neat!
SOLUTION KEY
1. (10 points) Use Cramer
MA 405-002 Spring 2014
Jacob Norton
Quiz #7 - March 7, 2014
Answer the questions in the space provided on the page. Show all your work; the correct answer without any
work receives no credit. Keep your work neat!
SOLUTION KEY
1. (10 points) Determine whet
MA 405-002 Spring 2014
Jacob Norton
Quiz #3 - January 24, 2014
Answer the questions in the space provided on the page. Show all your work; the correct answer without any
work receives no credit. Keep your work neat!
SOLUTION KEY
1. (10 points) Consider th
MA 405 Introduction to Linear Algebra and Matrices
Quiz 3
February 5, 2010
Instructions: Show all work relevant to the solution of each problem. i.e. no credit will be given for just
the answers. There are two problems which carry a total 10 points. Good
MA 405 Introduction to Linear Algebra and Matrices
Quiz 2 Solution
January 29, 2010
Instructions: Show all work relevant to the solution of each problem. i.e. no credit will be given for just
the answers. There are two problems which carry a total 10 poin
MA 405 Introduction to Linear Algebra and Matrices
Midterm Exam 1
February 10, 2010
Instructions: Show all work and justify all your steps relevant to the solution of each problem. No texts,
notes, or other aids are permitted. Calculators are not permitte
To see why .4 = PDF' 1 note that since lite vi are eigenvectore of]
.-1F=.dcfw_1r. 1r! I'll
= 1': sh .Jh . . Av
=cfw_Alr'l 31": Have
and by maIIiJr. multiplication
JE'E-'=cfw_1Fl t-.
a.
= i3: VI A: I"r:
Me Ill Mathematics 1|
Since AP = P5 then A = FDP]
I5 I IIS'IIJT IllE+
systems ui'linear equatinns is.
Ex: ,1]: :1 = cfw_1: E: I
In Gauss-Jordan Eimnerien, the augmeniaf Hermit msea'med aid: the system qr
linear museum is netsmtefly row retted ta missed ramesiaann hr.-
Shame;- a I in ens-y e
cfw_themes '3
eigenvectcrs he vl, v1," r, v _ New let P be the matrix 1iII.rhI:nse columns are These eigenvectcrs,
i_e_
P: [vE v! vIL ] _
Then
.4 = PEP] [er eeluivalentljlpr P".-lP = B":
.11 D
n
nemn
D: _
n I] AL.
JI" =[FDP"cfw_PDP"]~I[PDP"]
= ijr'e1ncfw_P-'P. .nF'
= P
Since 4-H" = f we have
or. +51: = I y. + an = [I
_ and
or. Hit-1 =l.l LT. +.rf;l.'1 =I _
or equivalently
Both systems of equations have the same coefcient matrix, i,e_
Ia b]
.c d_ _
The augmented matrices for these systems are
a it | ] In I: II]-
and
[e d
The intieutitgpr [remix has the property that for an]; square matrix. of order :1, H,
" L-i = J = :1
I is the onlyr mall-is That satises this property.
El Inverse Matrices
Err-en these-are men-i1- :t. cfw_filters exists a Ef! Heurissue that
J3 = Hal = I
I
cfw_Ali-)4 =cfw_J-I]r
- -l _l 1.
e24: t r
cfw_3" =B".-t
cfw_fl-I =cfw_JI'IJ' where rEEI
E.Ia.t_nE|e 1': Conrm that [.43] ' = 43".! holds for the matrices
[j 3 7 J-
Linuu' Algebra [W] - Leanne Mates
The idea of a matrix inverse can be related to The preble
er, in matlist fcnn
Dr
in an eigenvalue prehlem- Fnr the nub-is A as given ahnve it turns nut that the ranking
vectnr is
r=cfw__l4 cfw_LIE [LEE [LET H-213)?
Linear Algebra [V] - Lcctur: utes
and sun lite sites would be listed in the erder 14,11.-
Elf ceur
MA 405 Introduction to Linear Algebra and Matrices
Final Exam
May 10, 2010
Instructions: Show all work and justify all your steps relevant to the solution of each problem. No texts,
notes, or other aids are permitted. Calculators are not permitted. Please
MA 405 Introduction to Linear Algebra and Matrices
Quiz 1 Solution
January 22, 2010
Instructions: Show all work relevant to the solution of each problem. i.e. no credit will be given for just
the answers. There are two problems which carry a total 10 poin
MA 405 Introduction to Linear Algebra and Matrices
Midterm Exam 3 Solution
April 23, 2010
Instructions: Show all work and justify all your steps relevant to the solution of each problem. No texts,
notes, or other aids are permitted. Calculators are not pe
MA 405 Introduction to Linear Algebra and Matrices
Midterm Exam 2
March 13, 2010
Instructions: Show all work and justify all your steps relevant to the solution of each problem. No texts,
notes, or other aids are permitted. Calculators are not permitted.
MA 405-002 Spring 2014
Jacob Norton
Quiz #6 - February 28, 2014
Answer the questions in the space provided on the page. Show all your work; the correct answer without any
work receives no credit. Keep your work neat!
SOLUTION KEY
1. (NO points) (This quiz
MA 405-002 Spring 2014
Jacob Norton
Quiz #8 - March 19, 2014
Answer the questions in the space provided on the page. Show all your work; the correct answer without any
work receives no credit. Keep your work neat!
SOLUTION KEY
1. (10 points) Consider the
MA 405-002 Spring 2014
Jacob Norton
Quiz #9 - April 2, 2014
Answer the questions in the space provided on the page. Show all your work; the correct answer without any
work receives no credit. Keep your work neat!
SOLUTION KEY
1. (10 points) Is the followi
MA 405 - 002: Introduction to Linear Algebra
Homework 7
November 6
Instructions: You are encouraged to collaborate with other students. However you must write
your own solutions and list the names of collaborators. Explain the steps of your solution clear
is no entry in. row 3, column I-
1:?
From the denitions of matrix addition and seals: multiplication the Following general
properties can be shown-
ee-rm roperiies of Senior Multiplication :11:de Addition
Lem, B and C be men-ices oj'iiie same size. I'll-e
we have to row reduce the coefcient matrix to reduced row echelon form- "Thus there is no
real benet to solving a system of 31 equalions in e variables using a matrix inverse- However
the idea does highlight The following connections-
Mathl Ill - Mathemat
In matrix notation this system is
'l 2 _. -
i s"
The eigenvalues of [1 I] turn out to be l=i and .1: :3 with associated
1=C]e"[ I ]+C1e3'[l]_
1 L .l
From the initial coalitions we have
I 'I
eigenvectors v1 =[ ] and v1 =[ ]_ Thus, from cfw_I the general so
Methl Ill) Mathematics II
A linear equation in more Item :1 1I.-'ariahles cfw_where 1:33] is said to represent an :1-
di-meosional hyperplane"- The geometric interpretation cfw_in terms of rows of the solution to a
system of linear equations involving suc
1" 39+ 5: = 1 is equivalent to 31+3-p + |5_- =45
x+EIt-'+4s=7 1-+5p+4:=?
The following systems of linear equations are equivalent because we have just
subtracted the rst equation 'orn the second equation. (is. added -I times Ihe rst
equation to Ihe second