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
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
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
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 a
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 )
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 l
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 homogen
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 variab
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
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 a
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
.
.
. . = .
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 su
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
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
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 =
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
mu
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. (T
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.
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 m
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)
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. (T
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COURSE OUTLINE - MATH 213 Honors Linear Algebra I L01 - Instructor: M. Gree