MA122 Mock Midterm
Name:
Time Allowed: 80 minutes
Total Value: 65 marks
Number of Pages: 6
Instructions:
Cheat Sheet:
One 8:5" 11" page of study notes (both sides) is allowed as a reference
while completing the mock test. Please note, that the cheat sheet
Clicker Question 1 (Sept. 17)
Which one 13 a, vector equation for the line through the point P(1, 1) with
. . > 1
threat-1011 vector (1 : _1 E §1-2i
Addition:
Definition (Rn)
X1 +y1
Xn+Yn
Scalar Multiplication:
t
tX1
rxn Theorem For all W5)??? 6
MA122 Mock Midterm
Answers
(Full Solutions will NOT be posted;
use the MAC drop-in help centre if you have any questions.)
s
* Please remember that the mock test was meant as a means of providing an extra set of practice
questions and basis for a review c
Definition (Dot Product)
The dot product of vectors 2' = f , )7 = f in R is Theorem
Let if}? E R and t E R. Then
0 2-220, and 2.2:0 ifand only ir2:6,
a)? i7:i7-i<:
e 2* 0742):)? in)? z,
o (rm-n)? Wm? (n7)
Definition (Orthogonal)
The vectors 2'3?
3.2 Linear Mappings
1
2
A function that satisfies (L1) is said to preserve addition.
A function that satisfies (L2) is said to preserve scalar multiplication.
A function that satisfies both properties will preserve linear combinations: the
function is sai
To the linear system
311X1 + 312X2
a21X1 322X2
aml X1 + 3m2X2
we associate two matrices:
coeffICIent matrix
all 312 31m
321 322 32:1
3ml 3:112 3min
+ £31an : b1
+ £39an : b2
+ 3111an : bm
a ugmented matrix
311 312 31n
r321 322 32:1
aml 3:712 amn H
MA122A Introductory Linear Algebra
Spring Term, 2014
Instructor: Dr. Ping Zhang
Office Hours: Tuesdays & Thursdays 2:00 -3:00 p.m. (or other times by appointment)
Communication:
* All important course information will be posted on MyLearningSpace.
* E-mai
Directed Line Segments
We denote the directed line segment from P to Q by Pﬁ .
If 53 (j are the vectors corresponding to the points Pg Q (respectively), then
172
—r
Po:a—e
O XI
we may wish to treat the line segment QR as if it were the same as 0P.
2
Clicker Question 1 (Eloy. 20)
In which pair of vectors below. the two vectors are NOT orthogonal?
1 0 1 4
U —3 2 —3
A' 3 0 B' 3 2
{J —1 4 —1
0 1
0 —1 {a 4;, EN? 1}
o. _ D. :1: y 61113 —=—
0 1 HW‘H?” 2
0 —1 §1.4 Orthogonal Projections
a .L- I —’:. i '1
G
Applications to Euclidean vector spaces
Lemma
A set of vectors {i71. . . . . W} in R” is linearly independent if and only if the
rank of the coefficient matrix of the homogeneous system
t1l71+'“+tk\7k :0 l5
Theorem
If {71. . . . , Vk} is a linearly ind
Clicker QLIEStiOIl 1 (NIay. 13)
Which 0118 is NOT a subspace of R3?
331 £131 1 3 4
A. $2 332 :151 2 +tg 2 T {515152,t3 ER
333 $3
I$1=$2=$3€R:$120:$23>0-333>0}
H ]
0H2] [:ngHW}
{[1 Theorem
Let ‘31,. . W17} 6 R”. 13} can be written as a linear combination
1‘
If "' -'o ‘éé'1.r‘"'$ so *1: t- s. e:-
A vector v 111 R” 1s unlquelv determined bv1ts norn1 v and the amt vector
in its direction 6. In other ivorda given a direction and a real number 1“ Z 01
. —}- . . . . —}
there corresponds unlquelv a vector 1) 1
Clicker (211135131011
1
The cross product of the vectors ? : [ 2
1
3 —3
A. 0 B. 0 C. 0 2.1 Systems of Linear Equations
DEFINITION
An equation in a variables (unknowns) in, . . . , I” that can be written in the form,
called standard form,
ms] + -+anrn = b
§1.4 Projections The Perpendicular Part
For any vectors f, )7 E R, with a? at 6, dene the projection of )7 perpendicular to J?
to be
136er? 5 = )7 - ijz 37
perpj-(y) is perpendicular to f, and p11)ij ) + perpf (j!) = 5. Definition (Projection)
For vec
That is:
3:
(Expansion needs to be indicated by an arrow)
The arrow Technique is a
Example:
=
ij
ij
This matrix is called the
= (cof A)
or the cofactor matrix of A
(adjugate matrix)
= (cof A)
T
Let A =
. If det(A) = |A | = ad bc 0, then A is invertible an
MA122 Mock Final Exam
Name:
Time Allowed: 150 minutes
Total Value: 100 marks
Number of Pages: 8
Instructions:
Cheat Sheet:
One 8:5" 11" page of study notes (both sides) is allowed as a reference
while completing the mock test. Please note, that the cheat
MA122 Mock Midterm
Name:
Time Allowed: 80 minutes
Total Value: 65 marks
Number of Pages: 6
Instructions:
Non-programmable, non-graphing calculators are permitted. No other aids allowed.
Answer in the spaces provided.
Show all your work. Insu cient justica
§3.4
Theorem/Definition Let A be an m x n matrix. The set
S:{)_<R | Af:5}
of all solutions to a homogeneous system A)? : 5 is a subspace of R. It is
called the solution space of the system.
Definition (Nullspace)
The nullspace (or kernel) of a linear ma
^l
MA122
ock Final Exam
Time Allob ed: 1,lh11in11t( s
Total Value: 100!narks
Number of Pages: b
tructions:
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