MATH 102
Final Examination, Version 1, Answers
April, 13, 2016
Time: 2 hours
Chief Exam Administrator: E. Osmanagic
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Email:
@ualberta.ca
Please note that this exam will be marked electronically; BB pencil or dark ink
will give good scanni
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Student No.:
Lab 10 Solutions
Objective: To see that composition of two rotations about different coordinate axes is
equivalent to a single rotation about a fixed direction.
MATLAB Commands:
M=[1 2 3;4 5 6] Creates a certain 2x3 matrix and calls it
Name;
Student No.:
Lab 9 Solutions
Recall:
norm(v)
Returns the length of the vector v.
dot(a,b)
Takes dot product of two vectors a and b.
inv(A)
Inverse of matrix A.
det(A)
Computes the determinant of matrix A.
A' A-apostrophe: Computes the transpose of A
I 2. (15 points) For the given subsets, either prove that they are subspaces, 01' give a counter
\. \
example to Show that they are not.
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MLV: (OH6,1774% dHJ
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4 2 2
1. (10 points) Let A = 0 2 2 .
0 2 2
(8.) Find the characteristic polynomial and the eigenvalues of A?
(b) Find bwes for all the corresponding eigenspaces of A.
(c) Give an invertible P and diagonal D that satisfy A: PDP 1. (You must show both
the P
2. (8 points) Given below are equations for 2 planes in R3. Their intersection forms a line in R3.
10
2
3:31 w 2232
$1 + 233 + 4273
(it) Find this line and express it in parametric vector form.
(b) Find the point of intersection of the line from part. (a)
' 5. (20 points) Consider the set
X1 >? X m
W=cfw_(a+b+c,a c,2g*3c,b+c):a,b,cR.
(a) Prove that W is a subspace of R4.
a/ ? L 5 t (0 #0 PO) 000/ 2(0) 3(03 0 JFUM/
(94545 cfw_733 I 6W
AWFS, 2,:(g) 1/10, ,J' > ,
aiubfPacb" 4- (4d +16) 1' cfw_(6 "d " f3) 1 2(
3. (8 points) The sets 8 z cfw_1 +t+t2, 2132, t+t2 and c : cfw_1+t, 1+t2, 1713442 are two
bases for P2 (the space of all polynomials of degree at most 2).
(a) The standard basis for P2 is S = [1, Liz. Find the matrices P393 and ngc
(b) Find the change of
V 5. (5 points) Consider the set:
1 1
[3: b1: 3 ,b2= 0
1 1
(a) Show that B is an orthogonal set.
2
(b) Find the projection of u onto [IV = spancfw_b1, b2 where u 2: i 4 i.
W "1" 5
(c) What is u. ?
(d) Expand B to form an orthogonal basis 3" of R3.
(e)
1. (14 points) The sets B = cfw_1 + t + t2 , 2 t2 , t + t2 and C = cfw_1 + t, 1 + t2 , 1 t + t2 are two
bases for P2 (the space of all polynomials of degree at most 2).
(a) (4 points) The standard basis for P2 is S = cfw_1, t, t2 . Find the matrices PSB
MATHEMATICS 102
Winter 2017
Final Exam Review
The list below gives a short review of major topics and techniques on the material after the midterm.
Topics
Determinants, Cofactor Expansion, Row Reduction, Properties
Cross Product, scalar triple product
1. (14 points) The sets B = cfw_1 + t + t2 , 2 t2 , t + t2 and C = cfw_1 + t, 1 + t2 , 1 t + t2 are two
bases for P2 (the space of all polynomials of degree at most 2).
(a) The standard basis for P2 is S = cfw_1, t, t2 . Find the matrices PSB and PSC
(b
Math 102, Winter 2017
Assignment 2
Due: Friday, January 27, 23:55
Show your work / explain your reasoning! - Unsupported intermediate or final answers may result
in lost marks.
1. (18 points)
(a) Consider a triangle with vertices A, B and C. The line thro
1. (10 points)
(a) Find the point normal equation of the plane that passes through the point
P (0, 1, 1) and is perpendicular to the vector n = (1, 2, 1).
(b) Explain why the line y = (0, 1, 2) + t(1, 1, 1) cannot intersect the plane from part (a).
(c) Fi
Math 102, Winter 2017
Assignment 1
Due: Friday, January 20, 23:55
Show your work / explain your reasoning! - Unsupported intermediate or final
answers may result in lost marks.
1. In R5 , let
a = (1, 3, 1, 5, 0),
b = (4, 4, 9, 12, 6),
c = (4, 5, 0, 5, 3)
Math 102, Winter 2017
Assignment 6
Due: Friday, Mar 10, 23:55
Show your work / explain your reasoning! - Unsupported intermediate or final answers may result
in lost marks.
1. (20 points)
For the given vector spaces V and the given subsets W of V , either
1. A capital letter in the plane has the following position vectors for its corners:
(5, 10), (5, 10), (5, 8), (1, 8), (1, 0), (1, 0), (1, 8), (5, 8).
(a) On grid paper plot and identify the letter.
(b) The matrix transformation A :
R2
[2]
R2
is applied t
Math 102, Winter 2017
Assignment 4
Due: Monday, February 13, 23:55
Show your work / explain your reasoning! - Unsupported intermediate or final answers may result
in lost marks.
1. (10 points)
3 0
4 1
1 4 2
(a) Given A = 1 2 , B =
,C =
. Compute (BAT 2C)T
Math 102, Winter 2017
Assignment 8
Due: Friday, Mar 24, 23:55
Show your work / explain your reasoning! - Unsupported intermediate or final answers may result
in lost marks.
1. (10 points)
(a) Find an orthonormal basis for the subspace of R4 spanned by vec
MATHEMATICS 102
Winter 2017
INFORMATION FOR THE FINAL EXAM
DATE: Saturday, April 22, 2017
TIME: 09:00 , 2 Hours
LOCATION: PAVILION
Section
EQ1
EW1
ER1
ES1
EV1
EU1
Instructor
E. Osmanagic
E. Osmanagic
G. Tokarsky
E. Powell
E. Powell
G. Peschke
Rows
1,3,5
7
lvIath 102., Winter 2017 Assignmerfc 8
Duct Friday, Mar 2L 23:55
Shuw your mnk / (xplaill your cfw_vasoniugf , Ut15t1)p<)]f11(?(1 iutzolmxWham 01 final 2115me 111215? Ivmlli;
in iost 111211ks.
i. cfw_10 pointn)
(21.) Find an orthonormal Imsis fox" the snb
MATH 102
Midterm Examination, Version 1, Solutions
February, 10, 2017
Multiple Choice Questions
Each Multiple Choice question is worth 2 marks. Please enter your answer as a
CAPITAL LETTER in the BOX provided.
1. For a system (S) of five linear equations
1. Use cofactor expansion along a suitable row or column to compute det(A) if
2 10
0
6
0
2
3
1
A=
2
1 1
6
3
1
2 9
[10]
Winter 2017 - MATH 102 - Written Assignment #5 - due March 3 at 11:55 pm
Page 1 of 6
2. Use rules for computing with determinants to
MATH 102
Midterm Examination, Version 2, Solutions
February, 10, 2017
Multiple Choice Questions
Each Multiple Choice question is worth 2 marks. Please enter your answer as a
CAPITAL LETTER in the BOX provided.
1. Given three points P, Q, and R in R2 . If
Math 102, Winter 2017
Assignment 10
Due: Monday, April 10, 23:55
Show your work / explain your reasoning! - Unsupported intermediate or final answers may
result in lost marks.
1. (35 points) Consider the linear transformation T : R3 R3 given by reflection