MATA23H3 TERM TEST
Linear Algebra I a.
x "5
June 18 4. 5%,:
EXAMINER: B. McLellan
DURATION: 110 minutes
PART A: [20 TOTAL MARKS] MULTIPLE CHOICE: Circle one of fol
lowing (a)(e).
(1)[4 MARKS] Given the following statements:
I. If A, B 6 Mn are invertib
Fall 2016
Term Test 1 / L0101 Sample Solutions
CSC 165 H1
Question 1.
[8 marks]
Part (a) [2 marks]
Use a truth table to show that (p q) is logically equivalent to p q.
p
q
p q (p q)
q
p q
True True
True
False
False False
True False
False
True
True
True
Fa
MATA23 Lecture 1 Introduction to linear spaces, maps, systems

number line
1.5
R
0
Euclidean 1space
2
b

R
1
The plane: 2 lines intersect at origin
A point as an order pair (a,b)
Euclidean 2space
a
3

3 mutually lines meeting at the origin point in
MATA23 Lecture 4
Scalar Multiplication of Matrices
Let A=[ aij ] M n , k ( R ) let R .
The scalar product,
A , of the scalar
and matrix A is the matric
A=[ aij ]
Note:
A
1.
is the same size as A
A M n ,k ( R ) then A M n , k ( R )
2. If
Example
[ ]
4 7
le
MATA23 Lecture 6
Let
A
n k matrix. The linear system
be
n
vector b ( (R )
b =[3,0,3]
Ax=b
is consistent iff the
is in the span of the column vector A
in the span of
u=[0,1,1] ,
v =[1,1,0] ,
w =[1,1,1]
iff
b =x1 u + x 2 v + x3
w
[
][ ] [ ]
0 1 1 x 1
3
=
1
MATA23 Lecture 3
x 1+3 x 2=2
[] [] [ ]
2 x 1 +4 x2 =5
x 1 1 + x 2 3 = 2
2
4
5
Linear system
Column vector equation
b
A
x
[ ]
[]
1 3
2 4
x1
x2
The system can be written as
=
[ ]
2
5
A x =b , this is a matrix representation of
a linear system
A x
is equival
MATA23 Lecture 10
Example
Solve the system and the corresponding homogeneous system
x 1+ 2 x 2+ 4 x 3x 4 =0
x 2x 3 + x 4 =1
2 x 1x 2 +3 x 3 +3 x 4=5
x 1+3 x 2 +5 x3 2 x 4 =1
[
 ]
1 2
4 1 0 0 R R 2 R
3
1
0 1 1 1 0 1 3
R 4 R4 R1
2 1 3
3 0 5
1 3
5 2 0 1
[

MATA23 Lecture 8
A M n ,k ( R ) . Determine which
system A x =b is consistent.
n 1 matrices
If
A
is inevitable
If
A
is not square or not inevitable
works for all
b
are possible so that the
b R n
find
b
Example
Find all

b1 , b2 , b3
so that
cfw_
x + y +2
MATA23 Lecture 2
v =[ 1,2 ] ,
w =[ 2,1 ]
of
v
Any vector in
R
2
could be written as linear combination
w
and
Row vectors
[ 2,5 ] =s [ 1,2 ] +t [ 2,1 ] =4 [ 1,2 ] 3[2,1]
Column vectors
[ ] [] [] [] []
2 =s 1 +t 2 =4 1 3 2
5
2
1
2
1
v , u ,
w
Assume 3 no
MATA23 Lecture 5
The
n n
Identity matrix, denoted I or ( I n ) , is defined by
IA= A
Identity property means
Usually in a pattern of
I =[ ij ] ,
[ ]
1 0 0
1 0
I=
I = 0 1 0 etc .
0 1
0 0 1
If A= [ aij ] M n . n ( R ) , the element
[ ]
aij ,i=1, , n
are cal
MATA23 Lec09
If
W =sp ( w1 , w2 , , wk )
subspace of
is the span of
k
vectors in
Rn , the W is a
Rn
Proof
k >0
Case 1,
There is at least one
k
k
i=1
i=1
w W nonempty
Let u= i
wi , v = i
wi W
k
k
i=1
i=1
u +v = i
w i+ i
wi
( i
wi + i
wi)
k
i=1
( i
MATA 23H3
Tutorial Five
Summer 2013
For Tutorials on Tuesday June 11 and Wednesday June 12
Topics covered in this assignment are from part of Section 1.5 and part of Section 1.6 of the textbook.
Particularly, the following topics are covered:
Computation
University of Toronto at Scarborough
Department of Computer and Mathematical Sciences
Linear algebra I
MATA23 Winter 2016
Fraleigh & Beauregard, Pages 152  153
1
2
3
University of Toronto at Scarborough
Department of Computer and Mathematical Sciences
Linear algebra I
MATA23 Winter 2016
Fraleigh & Beauregard, Pages 153  154
34. First the inverse image of U is T1(U) = cfw_ x Rn  T (x ) U.
Since U is a subspace of Rm
University of Toronto at Scarborough
Department of Computer and Mathematical Sciences
Linear algebra I
MATA23 Winter 2016
Fraleigh & Beauregard, Pages 248  249
40.
1
Fraleigh & Beauregard, Pages 261  262
2
3
4
University of Toronto at Scarborough
Department of Computer and Mathematical Sciences
Linear algebra I
MATA23 Winter 2016
Fraleigh & Beauregard, Pages 271  273
1
2
3
University of Toronto at Scarborough
Department of Computer and Mathematical Sciences
Linear algebra I
MATA23 Winter 2016
Fraleigh & Beauregard, Pages 300  302
1
2
3
Fraleigh & Beauregard, Pages 315  316
=
4
5
6
Answer to the addition:
768 1280
1.
.
25
MATA 23H3
Tutorial Six
Summer 2013
For Tutorials on Tuesday June 25 and Wednesday June 26
Topics covered in this assignment are from part of Section 1.6 and part of Section 2.1 of the textbook.
Particularly, the following topics are covered:
Structure of
MATA 23H3
Tutorial Four
Summer 2013
For Tutorials on Tuesday June 4 and Wednesday June 5
Topics covered in this assignment are from Section 1.4 and part of Section 1.5 of the textbook. Particularly,
the following topics are covered:
c
Gauss Reduction of
MATA 23H3
Tutorial Nine
Summer 2013
For Tutorials on Tuesday July 16 and Wednesday July 17
Topics covered in this assignment are from Section 4.1, Section 4.2, and part of Section 4.3 (Pages 238  265)
of the textbook. Particularly, the following topics a
MATA 23H3
Tutorial Eight
Summer 2013
For Tutorials on Tuesday July 9 and Wednesday July 10
Topics covered in this assignment are from Section 2.3 and part of Section 2.4 (Pages 154159) of the textbook.
Particularly, the following topics are covered:
The
MATA 23H3
Tutorial Ten
Summer 2013
For Tutorials on Tuesday July 23 and Wednesday July 24
Topics covered in this assignment are from part of Section 4.3 (Pages 265  270) and part of Section 5.1 (up
to and including Theorem 5.1 in Page 296). Particularly,
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
Term Test
MATA23H
Linear Algebra I
Examiner: S. Chrysostomou
Date: Friday, March 9, 2012
Duration: 110 minutes
FAMILY NAME:
GIVEN NAMES:
STUDENT NUMBER:
DAY AND TIME OF YO
MATA23 Lecture 7
n n matrix A is invertible if there exists an
A
n n
matrix B such that
BA= AB=I . The matrix B is called the inverse of A and is denoted by
A1 . If A is not invertible, it is said to be singular
Example
[
] [ ]
A= 2 5 , B= 3 5
1 3
1 2
[
]