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 4I
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
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
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 abo
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
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:
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
maxi
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
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 s
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 feasi
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 dualit
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
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
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
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
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 : R
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 t
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 t
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.
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
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
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
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
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
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
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
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