MAT1503/101/3/2017
Tutorial letter 101/3/2017
Linear Algebra
MAT1503
Semesters 1 & 2
Department of Mathematical Sciences
IMPORTANT INFORMATION:
This tutorial letter contains important information about your
module.
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university
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Question 4
If z = r(cos + i sin ) is a complex number in polar form then from
De Moivres theorem we
z n = (r(cos + i sin )n = r n (cos n + i sin n)
for any natural number n.
for any natural number n by the binomial theorem we get
n n
n nk k
Question 3(c)
If n = (a, b, c) is a vector normal to a plane containing a point (x0 , y0 , z0 ). If
(x, y, z) is any point in the plane then the equation of the plane is given by
n (x x0 , y y0 , z z0 ) = 0
a(x x0 ) + b(y y0 ) + c(z z0 ) = 0
Example:
If u
Question
4
Let C =
3
2(c)
1
then
2
det(C) = (4)(2) (3)(1) = 8 3 = 5
since the det(C) 6= 0 it follows that C 1 has an inverse
The inverse is given by:
1 2 1
2/5 1/5
1
C =
=
3/5 4/5
5 3 4
Finding the determinant of the inverse of C we get
det(C 1 )
2
4
1
Chapter 13. VECTORS IN THREE DIMENSIONAL SPACE
Lets begin with some names and notation for things:
R is the set (collection) of real numbers. We write x R to mean that x is a
real number. A real number is also called a scalar because it can be used to s
MAT1503/203/2/2015
Tutorial Letter 203/2/2015
LINEAR ALGEBRA
MAT1503
Semester 2
Department of Mathematical Sciences
IMPORTANT INFORMATION:
This tutorial letter contains solutions
to assignment 03.
BAR CODE
Learn without limits.
university
of south africa
MAT1503/202/2/2015
Tutorial Letter 202/2/2015
LINEAR ALGEBRA
MAT1503
Semester 2
Department of Mathematical Sciences
This tutorial letter contains solutions
for assignment 02.
BAR CODE
Learn without limits.
university
of south africa
QUESTION 1
Let u = (u1
RATIONALIZING
A student in the group just notified me of the circled portion below in the document I uploaded (which
he did) and asked me if the second part (with question mark) is needed. Actually, he called it a mistake
but this is not a mistake. It is
Question 2(d)
If A is an n n matrix and k is any scalar then
det(kA) = k n det(A)
Example:
If
then
3 2
D=
1 1
6 4
2D =
2 2
and
det(2D) = 12 8 = 20
Thus
det(D) =
20
det(2D)
=
= 5
2
2
4
1
Question 1(b)
To verify that x = 19t 35, y = 25 13t, z = t t R is a solution of
the system
2x + 3y + z = 5
5x + 7y 4z = 0
Substitution of solution into first equation of the system we get:
LHS
= 2x + 3y + z = 2(19t 35) + 3(25 13t) + t
= 38t 70 + 75 39t +
Question
2(a)
cos sin
,
a)i)If A =
sin cos
find det(A) :
det(A) = cos cos + sin sin = cos2 + sin2 = 1
a+1
a
,
a)ii) If A =
a
a1
find det(A) :
det(A) = (a + 1)(a 1) a2 = a2 1 a2 = 1
b) Using co-factor expansion to find det(B) where
3 0 0 0
5 1 2 0
B=
2
Question 3(a)
If u, v R3 then
ku vk is the area of the parallelogram determined by vectors u and v
Example:
If u = (2, 1, 0) and v = (0, 1, 2) then
uv
i
= 2
0
1
= i
1
j k
1 0
1 2
2 0
2 1
0
+k
j
0 1
2
0 2
= 2i 4j + 2k
The area of the parallelogram is
Question 1(a)
The elementary row operations on a matrix are given by:
Multiply a row of a matrix by a nonzero constant: aRi Ri
Interchange any two rows of a matrix: Ri Rj
Add a multiple of one row to another row of a matrix: aRi + Rj Rj
1
Question 3(e)
Find an equation of plane that is perpendicular to the vector u = (2, 1, 0)
and passing through the terminal point of vector u assuming that the vector
is in standard position
Taking the normal n as the vector u. The plane passes through the
Question 3(d)
Two planes with normals n1 and n2 are parallel n1 = n2 for some
scalar .
Example:
Find an equation of plane that is parallel to plane
2x 4y + 2z = 0
and passing through the point (2, 2, 1).
Extracting the normal vector of the given plane we
How To Remember Special Values of Sine and Cosine
The following is a special table for remembering the special exact values of
the sine and cosine functions in Quadrant I. The key to the following table is
just knowing a few simple patterns. The first is
MAT1503/203/1/2017
Tutorial letter 203/1/2017
LINEAR ALGEBRA
MAT1503
Semester 1
Department of Mathematical Sciences
This tutorial letter contains solutions for assignment 03.
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university
of south africa
SEMESTER 1
ASSIGNMENT 03
Fix
MAT1503/102/1/2017
Tutorial letter 102/1/2017
LINEAR ALGEBRA
MAT1503
Semester 1
Department of Mathematical Sciences
This tutorial letter contains additional to the Study Guide
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CHAPTER 4
COMPLEX NUMBERS
4
UNIVERSiTY EXAMINATIONS UNIVERSITEITSEKSAMENS
N I A =
U S mum
MAT1 503 OctoberfNovember 2014
LINEAR ALGEBRA
Duration 2 Hours 100 Marks
EXAMINERS :
FIRST DR L GODLOZA DR ZE MPONO .
Closed book examination
This examination question pager remains the pro
Jim Lambers
MAT 169
Fall Semester 2009-10
Lecture 28 Notes
These notes correspond to Section 10.5 in the text.
Equations of Planes
Previously, we learned how to describe lines using various types of equations. Now, we will do the
same with planes. Suppose
A Matrix is an array of numbers:
A Matrix
(This one has 2 Rows and 3 Columns)
To multiply a matrix by a single number is easy
These are the calculations:
24=8 20=0
21=2 2-9=-18
We call the number ("2" in this case) a scalar, so this is called "scalar mult
3
1. Express the complex number 1 + i
Z=
1+i
r= |z| =
3
3
, x =1 and y=
x2 + y 2
=
1 +( 3)
2
2
1
= sin =
and cos
2
3
2
in polar form.
=2
3
then =
3
1+i
3
3
= r(cos +isin =2 cos +isin )
( aib )2
2. Let a and b be real numbers such that
(i)
Prove that b=
Students Number _
Type Only Answers
Given A =
2 4
1 3
Perform the following row operations beginning with matrix A and using your answer to each problem as the
matrix for the next.
1) 2R2 + R1 R1
4) Given the matrix
2) R1 R2
1 6 5
2 3 1
0 2 4
3)
repre
MAT1503/202/1/2016
Tutorial Letter 202/1/2016
LINEAR ALGEBRA
MAT1503
Semester 1
Department of Mathematical Sciences
This tutorial letter contains solutions
for assignment 02.
BAR CODE
Learn without limits.
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of south africa
SEMESTER 1
SOLUTIONS T
MAT1503/202/2/2016
Tutorial Letter 202/2/2016
LINEAR ALGEBRA
MAT1503
Semester 2
Department of Mathematical Sciences
This tutorial letter contains solutions
for assignment 02.
BAR CODE
Define tomorrow.
university
of south africa
SEMESTER 1
SOLUTIONS TO ASS
MAT1503/203/2/2016
Tutorial Letter 203/2/2016
LINEAR ALGEBRA
MAT1503
Semester 2
Department of Mathematical Sciences
This tutorial letter contains solutions
for assignment 03.
BAR CODE
Define tomorrow.
university
of south africa
SEMESTER 1
ASSIGNMENT 03
Fi
MAT1503/201/2/2016
Tutorial Letter 201/2/2016
LINEAR ALGEBRA
MAT1503
Semester 2
Department of Mathematical Sciences
This tutorial letter contains solutions
for assignment 01.
BAR CODE
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of south africa
SEMESTER 1
SOLUTIONS TO ASS
Two matrices are equal if and only if
The order of the matrices are the same
The corresponding elements of the matrices are the same
Addition
Order of the matrices must be the same
Add corresponding elements together
Matrix addition is commutative
Matrix
Question (b)
The cancellation law for arithmetic: If ab = bc and a 6= 0 then b = c
does not hold in general, for matrix multiplication.
Consider
matrices
0 1
,
A=
0 2
1 1
,
B=
3 4
2 5
C=
3 4
3 4
3 4
and AC =
then AB =
6 8
6 8
Although A 6= 0 canceling A
Question (i)
If A and B are square matrices of order n then det(AB) = det(A) det(B)
The n n matrix A is invertible det(A) 6= 0
If A is an invertible matrix then det(A) 6= 0 and we get that there exists a
matrix A1 such that AA1 = I.
Taking the determina
Question (c)
A square matrix A is invertible det(A) 6= 0
If A is an n n matrix that contains a row of zeros, suppose that aij = 0
for all j = 1, 2, n then finding the determinant by a co-factor expansion
along row i we get
det(A)
=
Pn
=
Pn
j=1 aij Cij
j=