PRACTICE FINAL
This practice final is not representative of Drurys final exam.
1.
A is a 4 x 4 non-diagonalizable matrix with two unique eigenvalues. How many
distinct possible forms for its Jordan canonical matrix are there (the eigenvalues
are placed on
Tutorial 4:
Orthogonalization
What we will learn:
- Inner products
- Projections on lines
- Projections on subspaces
- Least squares approximation
- Orthogonal bases
- Gram-Schmidt orthogonalization
- QR decomposition
What you should understand, but will
Tutorial 5:
Determinants
What we will learn:
- Pivot method for determinants
- Cofactor method for determinants
- Properties of determinants
- Cramers rule
- Inverses
What you should understand, but will not be emphasized here:
- Permutation method for de
Tutorial 6:
Midterm Exam Review
We will cover examples of 5 concepts:
- LU and LDU decomposition
- Column space and nullspace of matrix
- Gram-Schmidt and QR decomposition
- Determinants
- Transition matrix and change of coordinates
LDU Decomposition
Find
Tutorial 3:
Vector Spaces & Subspaces
What is a vector space?
A vector space V is a set of elements, called vectors, given
two operations, called vector addition and scalar
multiplication, that satisfy eight rules, called axioms.
The way the two operation
Tutorial 6:
Linear Transformations
What we will learn:
- Coordinate vector
- Change of basis matrix
- Linear transformation
- Matrix representation of transformations
- Matrix representation of reflections about planes
- Kernel & image of transformations
Tutorial 10:
Fourier Series
What we will learn:
- Fourier series
- Discrete Fourier Transform (DFT)
- Fast Fourier Transform (FFT)
What is a Fourier series?
The Fourier series of a periodic function f is the
representation of f as an infinite sum of sines
Tutorial 9:
Singular Value Decomposition
What we will learn:
- Similar matrices
- Jordan canonical form
- Singular value decomposition (SVD)
- Perron-Frobenius theorem
- Markov chains
What is a similar matrix?
A matrix A and B are similar, if for some inv
Tutorial 8:
Eigenvalues & Eigenvectors
What we will learn:
- Eigenvalues
- Eigenvectors
- Diagonalization
- Multiplicities
- Normal matrices
- Spectral theorem
What are the eigenvectors and eigenvalues of a matrix?
The eigenvectors of a matrix are the vec
McGill University
(Faculty of Science)
Final Examination
MATH-270: APPLIED LINEAR ALGEBRA
Date : Dec. 16, 2014
Examiner : J.J. Xu ;
Time : 2:00pm 5:00pm.
Associate Examiner : Axel Hundemer
INSTRUCTIONS
1. Write your name and student number on this examina
McGill University
Department of Mathematics & Statistics
MATH 270 Midterm Test
Version 1
50 minutes duration
Only faculty standard calculators allowed
This examination has 8 multiple choice questions. They are to be answered on the answer card provided.
B
Tutorial 4:
Orthogonalization
What we will learn:
- Inner products
- Projections on lines
- Projections on subspaces
- Least squares approximation
- Orthogonal bases
- Gram-Schmidt orthogonalization
- QR decomposition
What you should understand, but will
Tutorial 2:
LU Decomposition
What is LU Decomposition?
A matrix A can be factored into two matrices, a lower
matrix L and an upper matrix U.
=
A lower matrix has 0s above the diagonal and 1s on the
diagonal.
1
0 0
= 21 1 0
31 23 1
A upper matrix has 0s
Final Examination
April 18, 1997
MARKS
8 1. a Given
Mathematics 189-270B
8
x , x = sin3x + sin33x + sin35x + ; 0 x
1
3
5
1
X 1
obtain a numerical value for
.
n=1 n
6
10
2
b Solve the initial-boundary value problem: 12 @ = @ ; t 0; 0 x
@t @x2
0; t = 0; ;
Chapter 5
LINEAR VECTOR SPACE
1.
Denition of a Vector Space
After have studied the geometric vectors in Rn space, we turn to study
the general linear vector space, by extending all the concepts and notations dened in Rn space. We start with considering a
Chapter 4
THE VECTOR SPACE RN
In last chapter, it is seen that every vector in 3D space can be considered as a triples of ordered numbers, and described in the matrix
form: = [u1 , u2 , u3 ]T under a basis (e1 , e2 , e2 ) is introduced in R3 .
u
The conce
Chapter 1
REVIEW OF BASIC LINEAR ALGEBRA:
MATRICES AND LINEAR EQUATIONS
1.
Matrices
Suppose given two matrices A = [aij ] and B = [bij ]. The two matrices
are equal,
A = B,
if and only if
They have the same dimensions;
All corresponding elements are equal
122
APPLIED LINEAR ALGEBRA
Due to that u (S) , we have u, wi = 0(i = 1, k). Thus, we may
derive
ai u, wi = 0,
u, v =
i=1k
which implies that
u span(S) ,
(S) span(S) .
By combining the above results, we conclude that
(S) = span(S) .
5.
Mappings
We now are
145
LINEAR VECTOR SPACE
Solution: Relative to the standard basis, we have matrix
0 1 0
AT = 1 0 1 .
1 1 1
Furthermore, the change-of-basis matrix Q has the columns equal to the
standard coordinate vectors of the vectors of C. Hence, we have
1 1 1
Q = 0 1
McGill University
Math 270-2014 Fall: Applied Linear Algebra
Written Assignment 1: due on Tuesday, Sept. 30, 2014, hand-in in class
(Total 80 point)
1. (10 points) Given the following invertible matrix
[
]
3 1
A=
.
1 0
Express A in the form of product of
McGill University
Math 270-2014 Fall: Applied Linear Algebra
Written Assignment 2: due on Tuesday, Oct. 21, 2014, hand-in in class
(Total 60 point)
1. (10 points) Assume that A has eigenvalues 1 and 2 with corresponding eigenvectors X1 and
X2 .
(a) Find A
McGill University
Math 270-2014 Fall: Applied Linear Algebra
Written Assignment 4: due on Tuesday, Nov. 18, 2014, hand-in in class
(Total 80 point)
(*) Late handing-in is not acceptable!
1. (10 points) Let V = cfw_v R1 |v > 0. Show that V is a vector spac
McGill University
Math 270-2014 Fall: Applied Linear Algebra
Written Assignment 2: due on Thursday, Nov. 6, 2014, hand-in in class
(Total 80 point)
(*) Late handing-in is not acceptable!
1. (*10 points) Let A be an (m n) matrix.
(a) If U is an invertible