MAT343, Section 1, Fall 2014
TEST 1 - SOLUTIONS
Name: (BLOCK LETTERS PLEASE!)
Do NOT write your answers on this sheet please!
Scale:
30 : A
25 : B
20 : C
(15A,18B,18C out of 64)
Problem 1: Compute the intersection of the three planes in R3 , given by t
MATLAB sessions: Laboratory 1
1
MAT 343 Laboratory 1
Matrix and Vector Computations in MATLAB
In this laboratory session we will learn how to
1. Create matrices and vectors.
2. Manipulate matrices and create matrices of special types
3. Add and multiply m
Zizhao Huang, MAT343 12:00-1:15PM
MATLAB Lab assignment #5
Name: Zizhao Huang
ASU ID Number: 1208174808
Class time: Tu & Thur 12;00PM - 1:15PM
Lab time: Thur 1:30PM 2:20PM
% Question 1a
% Based on the distribution of the points, I expect the best straight
Properties of determinants
School of Mathematical & Statistical Sciences
Determinants via row/column operations
Row operations aect determinants as follows:
A row swap changes the sign of a determinant
A
r r
i k
B
=
A
A row scaling multiplies a determin
1.
The 3rd row, 4th column entry of a matrix is below and to the right of the 2nd
row, 3rd column entry. T
2.
A matrix with dimensions mn, where m > n, has fewer rows than columns. F
3.
The ith row, jth column entry of a square matrix, where i > j, is cal
MatLab Lab Assignment 5
Exercises:
1) Exercise 1
a. Having an even distribution of data points both greater than and less than zero, I
would expect the best-fit line to be near perfectly horizontal.
b. Generated data
clf
format short e
x = [1:1:100]';
y =
MatLab Lab Assignment 4
1) Exercise 1
clf
T = [-0.5,0,0.5,-0.5;-1,1,-1,-1];
plot(T(1,:),T(2,:),'linewidth',2)
hold on
R = [0,1;1,0];
RT = R*T;
plot(RT(1,:),RT(2,:),'-r','linewidth',2)
hold on
grid on
axis equal
title('Reflection About The Line at 45 Degre
MATLAB Lab Assignment 1
Exercises:
1. Enter the matrices
A=[2,6;3,9], B=[1,2;3,4], C=[-5,5;5,3]
A =
2
6
3
9
B =
1
2
3
4
C =
-5
5
5
3
2. Matrix operations
a) A+B, B+A: Adding A+B and B+A yeilds the same results. Therefore, matrix addition is
commutative
an
As the value of n increases, the time to complete the solution using the MatLab fucntion inv(A)
increases greater than that of the A\b method.
The accuracy of A\b increases as n get larger, eclipsing that of inv(A).
2) Consider the 100X100 matrix:
n = 100
Loops and Files
Keeping a Running Total
60 5.8 Sentinels
5.9 Focus on Software Engineering:
Operators
5.2 Introduction to Loops:
The while Loop Deciding Which Loop to Use
5.3 Using the while Loop for Input gfd ONested Loops
Validation si
MATH 54 QUIZ 7 SOLUTIONS
MAR 19, 2014
GSI: MINSEON SHIN
(Last edited March 20, 2014 at 3:45pm.)
a b
3
Problem 1. Find an invertible matrix P and a matrix C of the form
such that A =
b a
2
1
A = P CP .
5
has the form
5
Solution. The characteristic polynomi
Math 3105 Test 1 September 26, 2012
Print your name and your section number and Sign below, and read the instructions.
Do not open the test until you are told to do so.
Name (printed):
Section: I:
Signature: [:2
This test has 11 questions on 7 pages. The
ARIZONA STATE UNIVERSITY
School of Mathematical & Statistical Sciences Definition
The nullspace of a matrix A E Rm" is the set
N(A) = cfw_x E R s.t. Ax = O
of all vectors solutions of the homogeneous system Ax = 0.
o N(A) is a subspace of Euclidean (Rn, +
Vector Spaces
School of Mathematical & Statistical Sciences
Sets and operations on sets
A set V is a collection of objects
(numbers, arrays, matrices, functions, pictures, foods, names, . . . )
Objects in the set V can be manipulated using 2 operations
Th
Elementary matrices and the LU factorization J
ARIZONA STATE UNIVERSITY
School of Mathematical & Statistical Sciences Elementary matrices
Definition
An elementary matrix is a matrix that differs from the identity matrix by a
single elementary row operat
Row Echelon Forms
School of Mathematical & Statistical Sciences
Row Echelon Form (REF)
Denition
An augmented matrix is in Row Echelon Form if and only if
1
The rst nonzero entry in each row (the pivot) is a one
2
The pivot in each row comes after the pivo
Systems of linear equations
School of Mathematical & Statistical Sciences
System of Linear Equations
m linear equations in n unknowns
a11 x1
a21 x1
.
.
.
am1 x1
+
+
+
a12 x2
a22 x2
.
.
.
am2 x2
+ +
+ +
+ +
Typically m = n (square system)
x1 , x2 , . . . ,
Determinants
School of Mathematical & Statistical Sciences
What is a determinant?
The determinant det(A) of a square matrix A is a scalar value dened
to help us assess whether A is singular or not
if det(A) = 0 then A is singular
if det(A) = 0 then A is n
Quiz 3, Section 11
P1. Find the area of the triangle with vertices A(1, 1, 1), B(1, 1, 2) and C(0, 2, 3).
Solution: First we choose one of the points A, B, C to be the common vertex for being the starting
point of the two vectors that form the two sides o
AricaFoster
MAT343Week4WrittenHW
SupposeAisa3x5matrixwithrank2.
3
a.DoesasolutionofAx=bexistforanybin R ?Explainindetail.
Ax=bis consitant for any b R m whererank ( A )=m . Inthisexample,Rank(A)=2,butbis
3
in R and32sothisisinconsistentanddoesnotguarantee
Because the det(AB) = det(A)*det(B), and the fact that A*A -1= I, where I is the identity matrix and the
determinant det(I) = 1, it can be shown that det(A)*det(A -1) = det(A*A-1) = det(I) =1.
We assume that det(A)0 and can manipulate det(A)*det(A -1) = 1
For a matrix to be invertible, it must have a unique solution when multiplied by a vector. Because vector
x does not equal vector y, then in order for matrix A to be invertible, they must have unique solutions A
is multiplied with each of the vectors. How
Arica Foster
MAT 343
Week 5 Written Homework
Because the rank(A) = rank(A TA), the least square solution is unique if and only if rank(A)=n where n =
the number of columns of matrix A. If rank(A)=n then all of the columns of the matrix are linearly
indepe
MAT 343 MATLAB Lab 2
Name: Arica Foster
% Exercise 1, Part (a)
n=1000; % set n=1000
A=floor(10*rand(n); % generate n x n matrix
b=sum(A')';
% generate vector in R^n
z=ones(n,1); % generate vector in R^n
(Question 1) Z is the exact solution to the system A
Arica Foster
MAT 343
MatLab 5
Exercise 1, Part (a)
All of the x values are positive ranging from 1 to 100. The y values randomly generated where
those values have an equal chance of being positive or negative. The best fit line for these values
will be a
MAT 343
Week 3Written HW
Arica Foster
The only way matrix vectors are linearly independent is if there is no relation between them and
none can be represented as a combination of another.
The null space is the vector space that within the given subspace t