MATH 223: Notes on determinants.
Richard Anstee
We seek a determinant function det : Rnn R that satisfies various natural properties.
det I = 1
If B is obtained by multiplying row i of A by t then det(B) = t det(A)
If B is obtained from A by interchang
MATH 223. Orthonormal bases and Gram-Schmidt process.
Richard Anstee
Consider a vector space V with an inner product < , >: V V R. We are interested in
finding orthonormal bases for vector spaces. An orthonomal basis cfw_w1 , w2 , . . . , wt is a basis s
MATH 223
Assignment #6
due Friday October 28.
1. Determine bases for the following subspaces of R3 .
a) the line x = 5t, y = 2t, z = t has basis
5
2 .
1
b) all vectors of the form (a, b, c)T such that a 3b = 2c has the basis
3
2
0,1
0
1
2. Let
We hav
Last name:
Student no.:
Math 223 - Midterm 1 - Friday October 7, 2016 - Solutions
Please show your work. Correct final answers may not receive full credit if insufficient arguments
are given.
1. [20 marks] Using Gaussian Elimination, give all the solution
MATH 223
Assignment #7
due Friday Nov 4.
1. Let A be an m n matrix of rank 1. Show that there exist non zero vectors x Rm and y Rn
so that A = xyT . (Hint: Try a simple case. Also compute xyT for some simple choices for x, y.)
2. Let cfw_u1 , u2 , u3 be
MATH 223
Assignment #4
due Wednesday October 5 in class.
Extra office hours Tuesday October 4 from 5:10-6 in Math Annex 1118.
Midterm is scheduled for
Friday October 7. A practice midterm is posted.
x
0 0
1. i) det 10 x 0 = x2 (triangular matrx and det(A
MATH 223 Putnam Math Contest problem from 2013.
Problem. Let A be a 2013 2014 matrix of integer entries such that each row sum is 0 (i.e.
A1 = 0 where 1 is the 2014 1 vector of 1s and 0 is the 2013 1 vector of 0s. Show that
det(AAT ) = 2014k 2 for some in
Math 223, Section 101Homework #3
due Wednesday, September 28, 2011 at the beginning of class
Practice problems. Dont turn in these problems. Most of these problems are easy, and most of
them have answers in the back of the book, so they will be useful to
Math 223, Section 101Homework #5
due Friday, October 21, 2011 at the beginning of class
Practice problems. Dont turn in these problems. Most of these problems are easy, and most of
them have answers in the back of the book, so they will be useful to you t
MATH 223. Orthogonal Projections and Least Squares.
Richard Anstee
Orthogonal Projections
Let U be vector subspace of V . Then dim(U ) + dim(U ) = dim(V ). It is reasonable and
important in many problems to express a vector v V as a sum v = u + w where u
MATH 223. Quadratic Forms, Conic Sections.
Richard Anstee
2
2
When faced with a quadratic function such as x + 3xy + y + 2yz + 2z 2 we discover that we
can write it using a matrix:
1 3 0
x
x2 + 3xy + y 2 + 2yz + 2z 2 = [x y z] 0 1 2 y
0 0 2
z
and then
Math 223
Symmetric and Hermitian Matrices.
Richard Anstee
An n n matrix Q is orthogonal if QT = Q1 . The columns of Q would form an orthonormal
basis for Rn . The rows would also form an orthonormal basis for Rn .
A matrix A is symmetric if AT = A.
Theore
MATH 223: Linear Transformations and 2 2 matrices.
Richard Anstee
If we write
a b
A=
= [A(1) A(2) ],
c d
then
Ax =
a b
c d
x
= xA(1) + yA(2) .
y
We can consider functions
f : x Ax.
f (x) = Ax,
We note that Ax is a linear combination of the columns of A.
L
MATH 223: Some results for 2 2 matrices.
Richard Anstee
One can preview a lot of the theory in this course by looking at the special case of 2 2
matrices. The proofs are relatively easy in this limited context and not all the complexity is seen.
In partic
MATH 223: Diagonalization with Eigenvalues and Eigenvectors.
An application to bird populations (Leslie Matrix).
Sample computation
Let
"
.7 .3
2 0
A=
#
An application associated with this matrix is a simple model of a growing bird population. Let
xn = no
MATH 223: White and Blue Coordinates.
We intially understood that a vector ab was given in the standard way with ab = a 10 +b 01
and hence a units to the left and b units up from the origin.
We had an assignment question on assignment 1 changing b
MATH 223: Some results for 2 2 matrices.
Richard Anstee
Multiplicative Inverses
It would be nice to have a multiplicative inverse. That is given a matrix A, find the inverse
matrix A1 so that AA1 = A1 A = I. Such an inverse can be shown to be unique, if i
MATH 223
Different Levels of Generality
One challenge when doing Mathematics is to choose the right level of generality. In the context
of Linear Algebra there are at least three levels of generality. We began the course with 2 2
matrices
"
#
a b
A=
c d
a
MATH 223. Orthogonal Vector Spaces.
Let U, V be vector spaces with U V . We consider
U = cfw_v Rn : for all u U, < u, v >= 0
Theorem 0.1 U is a vector space.
Proof: We show that U is a vector space. Here we must verify that 0 U since this will not
follow
Math 223, Section 101Homework #8
due Friday, November 18, 2011 at the beginning of class
Practice problems. Dont turn in these problems. Most of these problems are easy, and most of
them have answers in the back of the book, so they will be useful to you
Math 223, Section 101Homework #7
due Friday, November 4, 2011 at the beginning of class
Practice problems. The usual rules apply.
1. I have linked to a web page with practice problems about complex numbers. You can go
to the URL http:/tinyurl.com/MATH223
Math 223, Section 101Solutions to Homework #8
due Friday, November 18, 2011 at the beginning of class
1. Friedberg, Insel, and Spence, Section 5.1, pp. 2578, #3(d) and #4(g)
3(d) (i) To nd the eigenvalues of A, we calculate its characteristic polynomial,
Math 223, Section 101Solutions to Homework #10
due Friday, December 2, 2011 at the beginning of class
1. Friedberg, Insel, and Spence, Section 6.2, p. 353, #2(f),(h). The term Fourier coefcient is
dened on page 348.
(f) Starting with the set S = cfw_w1 ,
Math 223, Section 101Solutions to Homework #4
due Friday, October 14, 2010 at the beginning of class
1. Friedberg, Insel, and Spence, Section 2.2, pp. 8485, #2(b),(e), #4, and #5(d)
#2(b) For clarity, let = cfw_e1 , e2 , e3 denote the standard ordered ba
Math 223, Section 101Solutions to Midterm #2
Wednesday, November 9, 2011
1. Label each of the following statements as either true or false. No justication is required; just
choose the correct answer.
(a) [1 pt] A linear transformation is invertible if and
Math 223, Section 101Solutions to Midterm #1
Wednesday, October 6, 2010
1. Label each of the following statements as either true or false. No justication is required; just
choose the correct answer.
[1 pt]
[1 pt]
[1 pt]
[1 pt]
ber c.
(e) [1 pt]
(a)
(b)
(c
Math 223, Section 101Solutions to Homework #3
due Wednesday, September 28, 2010 at the beginning of class
1. Friedberg, Insel, and Spence, Section 1.6, p. 54, #2(b),(d) and #3(b),(d)
#2. A set of vectors in Rn is linearly independent if every column of th
Math 223, Section 101Solutions to Homework #6
due Friday, October 28, 2011 at the beginning of class
1.
(a) Use elementary row and column operations (show your work) to transform the matrix
1 2
1 1 2
1
2
2
4
8
A=
3
6 4
2 3
4 8
2 6
5
to a matrix B of t
Math 223, Section 101Solutions to Homework #5
due Friday, October 21, 2010 at the beginning of class
1. Friedberg, Insel, and Spence, Section 2.4, p. 108, #17
Let V and W be nite-dimensional vector spaces and T : V W be an isomorphism. Let V0 be
a subspac