Department of Mathematics, University of Toronto
MAT223H1F - Linear Algebra I
Fall 2015
Tutorial Problems 3
1. Suppose a 3 3 matrix A has inverse
2
0
0
1
5
0
1
1 .
1
1
Find a matrix B such that (AB 4I3 )T A = C, where C = 0
0
2 5
0 1
2. Find all values of
Department of Mathematics, University of Toronto
MAT223H1F - Linear Algebra I
Fall 2014
Tutorial Problems 1
1. Suppose that the given matrix is the augmented matrix of a system of linear equations in the variables
x1 , x2 , x3 , x4 , x5 , x6 .
0 0 0 1 2 1
Department of Mathematics, University of Toronto
MAT223H1F - Linear Algebra I
Fall 2014
Tutorial Problems 1
1. Suppose that the given matrix is the augmented matrix of a system of linear equations in the variables
x1 , x2 , x3 , x4 , x5 , x6 .
0 0 0 1 2 1
Department of Mathematics, University of Toronto
MAT223H1F - Linear Algebra I
Fall 2014
Brief Course Description
Welcome to MAT223H1F Linear Algebra I. This sheet answers the most common questions about the course.
Please take a few minutes to read this c
Interpretation
for any z 0 with
!
i zi
= 1,
f (x) = max(aTi x + bi) z T (Ax + b)
i
for all x
this provides a lower bound on the optimal value of the PWL problem
min f (x) min z T (Ax + b)
x
x
" T
b z if AT z = 0
=
otherwise
the dual problem is to find
proof of equivalence of the dual problems (assume A is m n)
if u, v are feasible in (1), then z = v u is feasible in (2):
z1 =
m
!
|vi ui| 1T v + 1T u = 1
i=1
moreover the objective values are equal: bT z = bT (v u)
if z is feasible in (2), define vecto
Variants
LP with inequality and equality constraints
minimize cT x
subject to Ax b
Cx = d
maximize bT z dT y
subject to AT z + C T y + c = 0
z0
standard form LP
minimize cT x
subject to Ax = b
x0
maximize bT y
subject to AT y c
dual problems can be deriv
Outline
dual of an LP in inequality form
variants and examples
complementary slackness
Optimality conditions
primal and dual LP
minimize cT x
subject to Ax b
Cx = d
maximize bT z dT y
subject to AT z + C T y + c = 0
z0
optimality conditions: x and (y,
the alternative system has no solution because:
if t > 0, defining x
= w/t, z = u/t gives
z 0,
AT z + c = 0,
A
x b,
cT x
< bT z
this contradicts the lower bound property
if t = 0 and bT u < 0, u satisfies
u 0,
AT u = 0,
bT u < 0
this contradicts feasi
Dual infeasible problems
if d = then p = or p = +
proof: if dual is infeasible, then from page 53, there exists y such that
AT y 0,
cT y < 0
if the primal problem is feasible and x is any primal feasible point, then
AT (x + ty) b
for all t 0
therefore x +
Summary
PSfrag
p = +
d = +
p =
primal inf.
dual unb.
optimal
values equal
and attained
d finite
d =
p finite
exception
primal unb.
dual inf.
upper-right part of the table is excluded by weak duality
first column: proved on page 68
bottom row: proved
Strong duality
if primal and dual problems are feasible, then there exist x, z that satisfy
cT x = bT z ,
Ax b,
AT z + c = 0,
z 0
combined with the lower bound property, this implies that
x is primal optimal and z is dual optimal
the primal and dual opt
Duality theorem
notation
p is the primal optimal value; d is the dual optimal value
p = + if primal problem is infeasible; d = if dual is infeasible
p = if primal problem is unbounded; d = if dual is unbounded
duality theorem: if primal or dual problem
Department of Mathematics, University of Toronto
MAT223H1F - Linear Algebra I
Fall 2014
Tutorial Problems 2
0
1
1
1
1
3 1 0
1. Let W = Span , , . For what value of c is W ?
0
0 0 1
c
2
1
1
6
0
2 7
2. Let W = Span , , and S =
1 1
0
3
2
2
3 1
, .
Department of Mathematics, University of Toronto
MAT223H1F - Linear Algebra I
Fall 2014
Tutorial Problems 2
0
1
1
1
1
3 1 0
1. Let W = Span , , . For what value of c is W ?
0
0 0 1
c
2
1
1
0
1
1
1
0 1
1
3
Solution: We want the augmented matrix c
Department of Mathematics, University of Toronto
MAT223H1F - Linear Algebra I
Fall 2015
Tutorial Problems 5
1 (a) Express the following functions, if linear, as a matrix transformation:
x1
(i) T : R3 R dened by T x2 = x1 + 2x2 + x3
x3
|x1 |
x1
(ii) T :
Department of Mathematics, University of Toronto
MAT223H1F - Linear Algebra I
Fall 2015
Tutorial Problems 4
Key terms and ideas:
Here are the key terms and ideas we are trying to demonstrate in this tutorial question set. The page
number refers to the cou
Department of Mathematics, University of Toronto
MAT223H1F - Linear Algebra I
Fall 2015
Tutorial Problems 3
Key terms and ideas:
Here are the key terms and ideas we are trying to demonstrate in this tutorial question set. The page
number refers to the cou
Department of Mathematics, University of Toronto
MAT223H1F - Linear Algebra I
Fall 2015
Tutorial Problems 4
1
1. Let A = 2
1
4
5
3
3
4 . Express both A and A1 as a product of elementary matrices.
2
2. Let x and y be vectors in Rn . The inner (or dot) prod
Department of Mathematics, University of Toronto
MAT223H1F - Linear Algebra I
Fall 2015
Tutorial Problems 1
1. Consider the system of linear equations
x1 3x2 = 2
2x1 + 6x2 = 4
(a) Write the augmented matrix of the system, nd all solutions, and express you
Department of Mathematics, University of Toronto
MAT223H1F - Linear Algebra I
Fall 2014
Tutorial Problems 5
1. In class we have seen that matrix multiplication is not commutative. That is AB = BA even if both
products are dened (see Example 2 & 3, page 98
Department of Mathematics, University of Toronto
MAT223H1F - Linear Algebra I
Fall 2014
Tutorial Problems 4
2
1
1
1
1. Let T : R3 R2 be a linear transformation such that T 1 =
, and T 1 =
.
1
2
1
1
0
(a) Determine, if possible, T 1 .
1
0
2
3
(b)
Department of Mathematics, University of Toronto
MAT223H1F - Linear Algebra I
Fall 2014
Tutorial Problems 4
2
1
1
1
1. Let T : R3 R2 be a linear transformation such that T 1 =
, and T 1 =
.
1
2
1
1
0
(a) Determine, if possible, T 1 .
1
0
2
3
(b)
Friday October 17
START: 16:10
110 mins
University of Toronto
Department of Mathematics
MIDTERM EXAMINATION I
MAT223H1F
Linear Algbera I
EXAMINERS: P. Crooks, N. Jung, M. Mota, P. Samuelson, P. Sastry, S. Uppal
Last Name (PRINT):
Given Name(s) (PRINT):
St
Department of Mathematics, University of Toronto
MAT223H1F - Linear Algebra I
Fall 2014
Tutorial Problems 3
1
5
2
1(a) Is the set 1 , 3 , 2 linearly independent in R3 ?
1
2
1
2
1
5
1(b) Do the vectors 1, 3, and 2 span R3 ? (Note: It is possible to
Department of Mathematics, University of Toronto
MAT223H1F - Linear Algebra I
Suggestions for Review: Midterm Exam I
I suggest before you go through this document that you read through all your class notes and/or the textbook
and, as youre reading, solve
Department of Mathematics, University of Toronto
MAT223H1F - Linear Algebra I
Fall 2014
Tutorial Problems 3
1
5
2
1(a) Is the set 1 , 3 , 2 linearly independent in R3 ?
1
2
1
2
1
5
1(b) Do the vectors 1, 3, and 2 span R3 ? (Note: It is possible to