ASSIGNMENT 1 SOLUTIONS
(3g) We want to nd a, b and c and d such that ax + by + cz = d has general solution
(x, y, z) = (1 s, 2 3t, 4t 5s 1).
Substituting, we see that the equation
a(1 s) + b(2 3t) + c(4t 5s 1) = d
must be satised. Rerranging terms, we get
MATH 213 LAB 6 PRODUCTS OF SETS;
DIRECT SUMS OF VECTOR SPACES
1. Products of sets
Let X and Y be sets.
Denition 1. The (Cartesian) product of X and Y , written X Y , is the set of all ordered
pairs (x, y) where x X and y Y :
X Y = cfw_(x, y) : x X, y Y .
1. Linear (in)dependence
Let V be a vector space and let v1 , v2 , . . . , vn be a nite sequence of elements of V .
Denition 1.
Let x1 , x2 , . . . , xn be scalars. An identity of the form
(1)
0 = x1 v 1 + x2 v 2 + xn v n
is called a linear dependence re
1. Subspaces
Denition 1. Let V be a vector space and let W be a subset of V . We say that W is
closed under addition if v1 + v2 W for all v1 , v2 W .
closed under scalar multiplication if xv W for all x R and all v W .
a (vector) subspace of V if it is
1. Vector spaces
1.1. Addition. Let V be a set.
Denition 1. An addition law + on V is a rule that, to every pair (v1 , v2 ) of elements of
V , assigns a sum element v1 + v2 V to every pair (v1 , v2 ) where v1 , v2 V such that:
(1) (Associative law) (v1 +
1. Linear combinations and spans
Let V be a vector space.
Denition 1.
A sum of the form
x1 v 1 + x2 v 2 + + x n v n ,
where x1 , x2 , . . . , xn are scalars and v1 , v2 , . . . , vn V , is called a linear combination
of the vectors v1 , v2 , . . . , vn .
MATH 213 LAB 3 TEST 1 REVIEW
(1) For what pairs (a, b) does
x + ay
x + ay
bx + a2 y
+
+
+
a2 z
abz
a2 bz
= 1
= a
= a2 b
have no solution? A unique solution? Innitely many solutions?
(2) Find a homogeneous linear equation with general (parametric) solution
MATH 213 LAB 2
1. Introduction
The Gaussian elimination algorithm weve developed for transforming a matrix into reduced row echelon form allows us to solve any system of m linear equations in n variables.
An important issue that we havent touched on is th
MATH 213 LINEAR ALGEBRA EXERCISES
1. Linear equations
Terminology:
A linear equation is called degenerate if all of its coecients are equal to 0. (Its constant term need
not be 0.)
If k is a nonzero number, the linear equations
a1 x1 + an xn = b
and
k(a
MATH 213 LAB 1
1. Terminology
The leftmost nonzero entry in a row is called its leading entry. If this entry is 1 it is
called a leading 1.
A matrix is in row echelon form if
(1) All zero rows are at the bottom of the matrix.
(2) All nonzero rows have l
MATH 213 LAB 5 THE TRANSPOSE;
TECHNIQUES FOR PROVING THINGS ABOUT MATRICES
Denition: Let A = (aij ) be an m n matrix. Dene the
(i, j)-entry is aji :
T
a11 a12 a1n
a11
a21 a22 a2n
a12
.
.
. . = .
.
.
.
.
.
.
.
.
am1 am2 amn
a1n
transpose AT of A to
MATH 213 FALL 2015 HOMEWORK 3
DUE: FRIDAY, 13.11.2016
(1) Let n1 and n2 be positive integers. For i = 1, 2 and j = 1, 2, let Aij be an ni nj
matrix. Let
A11 A12
A=
A21 A22
(a) Find matrices X and Y such that
I 0
I Y
A
X I
0 I
(b) Find
I 0
X I
1
=
I Y
0 I
MATH 213 FALL 2015 SUPPLEMENTARY PROBLEMS
1. Problems
(1) Do the problems in Chapter 2 of Kuttler (the text-e-book) dealing with transpose, inverse and
elementary matrices.
(2) (a) Let m, n1 , n2 and p be positive integers. Let S 1 , S 2 , T1 and T2 have
MATH 213 LAB 9
A linear transformation is a function T : V W such that
(i) T (v + v ) = T (v) + T (v ), and
(ii) T (xv) = xT (v)
for all v, v V and all x R.
Let A Rnn . A nonzero vector v Rn1 is an eigenvector of A belonging to
the eigenvalue if Av = v.
MATH 213 ASSIGNMENT 4
DUE MONDAY, DECEMBER 7 IN CLASS.
(1) Let A be a 3 4 matrix and suppose that
1
N (A) = span 2 .
3
Let
A
.
A
Compute the dimensions of C(X), R(X), N (X) and N (X T ) for X = A, B, or C.
B= A A
and let
C=
(2) (a) Let u and v be no
MATH 213 ASSIGNMENT 3 SOLUTIONS
(1) (a) We assume that A11 is invertible. To kill the (2, 1)-entry of A, we perform the
block row operation subtract A21 A1 times row one from row two from by
11
I
0
multiplying A on the left by the block elementary matrix
MATH 213 ASSIGNMENT 2
DUE: FRIDAY 23/10/2015
(1) For 1 i, j 2, let eij = (eij ) be the 2 2 matrix such that
k
eij =
k
1
0
if i = k and j = ,
otherwise.
(a) Write down the matrices eij for all i, j. (That is, parse and understand the denition of eij .)
(b)
MATH 213 FALL 2015 TEST 2
INSTRUCTOR: MATTHEW GREENBERG
Solve two of problems (1)-(3) and solve two of problems (4)-(6). Write neatly in the booklet provided; write
your name and student number on it. Point values are indicated. You have 50 minutes. Good
MATH 213 REVIEW PROBLEMS
(1) Find the reduced row echelon form of A. Solve the equation Ax = 0.
1 2
3 9
(a) A = 2 1 1 8
3 0 1 3
1 2 3
4
5
7 11
(b) A = 1 3 5
1 0 1 2 6
(2) Find an invertible matrix U such that U A is in reduced row echelon form.
1 2 3
(a)
MATH 213 FALL 2015 TEST 1
INSTRUCTOR: MATTHEW GREENBERG
Solve all ve problems. Write neatly in the booklet provided; write your name and student
number on it. Point values are indicated. You have 50 minutes. Good luck!
(1) (a) 1 pt: Dene what it means for
MATH 213 TEST 3 REVIEW
(1) Let A be an m n matrix and let B be its reduced row echelon form.
(a) If rank A = r, then any r columns of A form a basis of C(A).
(b) If m > n then dim R(A) > dim C(A).
(c) If rank A = n then the rows of A form a basis of Rn .
MATH 213 ASSIGNMENT 2
DUE: FRIDAY 23/10/2015
(1) For 1 i, j 2, let eij = (eij ) be the 2 2 matrix such that
k
eij =
k
1
0
if i = k and j = ,
otherwise.
(a) Write down the matrices eij for all i, j. (That is, parse and understand the denition of eij .)
(b)
FACULTY OF SCIENCE
Department of Mathematics and Statistics
2500 University Drive NW
Calgary, AB, Canada T2N 1N4
ucalgary.ca
COURSE OUTLINE - MATH 213 Honors Linear Algebra I L01 - Instructor: M. Greenberg
Course: Honors Linear Algebra 1
LEC 1: MWF 9:00 9