meljun cortes matrix algebra rm104tr-10
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7/29/2019 MELJUN CORTES MATRIX ALGEBRA Rm104tr-10
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Lesson 10 - 1
Year 1
CS113/0401/v1
Matrix definition
Rectangular array of numbers
Size of matrix is given by no of
rows and no of columns e.g. A = 2 x 3 matrix
e.g. B = 3 x 3 matrix
2 9 16
1 0 -1
8 1 0
-1 0 1
4 1 5
LESSON 10MATRIX ALGEBRA
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Lesson 10 - 2
Year 1
CS113/0401/v1
Vectors A single row matrix is called a
row Vector
e.g. [ 5 9 1 2 ]
A single column matrix is called
column vector
e.g.16
1
0
-1
MATRIX ALGEBRA
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Lesson 10 - 3
Year 1
CS113/0401/v1
1 11 2
0 1 1
2 0 1
1 0 2
1 11 2
0 1 1
2 0 1
1 0 2
1+2 11+ 0 2+1
0+1 1+0 1+2
3 11 31 1 3
MATRIX OPERATION Matrix Addition
Must be of same dimension
result is of same dimension
E.g. A =
B =
A + B = +
=
=
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Lesson 10 - 4
Year 1
CS113/0401/v1
1 11 2
8 1 1
-2 0 1
1 9 -2
1 11 2
8 1 1
-2 0 1
1 9 -2
1+(-2) 11+ 0 2-1
8-1 1-9 1-(-2)
3 11 17 -8 3
MATRIX OPERATION
Matrix Subtraction Same rule as matrix addition
e.g.A =
B =
A - B = +
=
=
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Lesson 10 - 5
Year 1
CS113/0401/v1
5 2
1 -1
5 2
1 -1
5x2 2x21x2 -1x2
10 4
2 -2
MATRIX OPERATION
Matrix Multiplication Scalar Multiplication
e.g.A =
2A = 2
=
=
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Lesson 10 - 6
Year 1
CS113/0401/v1
MATRIX OPERATION
Matrix Multiplication No of columns is 1st matrix must
be equal no of rows in 2nd matrix
Result is of dimension No of rows in 1st matrix by no
of column in 2nd matrix
e.g. If A is of dimension 2 x 3
B is of dimension 3 x 1
Then R=A * B is defined
R is of dimension 2 x 1
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Lesson 10 - 7
Year 1
CS113/0401/v1
R1 C1
3 1
2 47 4
A =
B =8 0 5 43 2 11 1
AB =3 12 47 4
8 0 5 43 2 11 1
3x8+1x3 3x0+1x2 3x5+1x11 3x4+1x12x8+4x3 2x0+4x2 2x5+4x11 2x4+4x17x8+4x3 7x0+4x2 7x5+4x11 7x4+4x1
=
=
27 2 26 13
28 8 54 1268 8 79 32
MATRIX OPERATION
Matrix Multiplication
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Lesson 10 - 8
Year 1
CS113/0401/v1
In matrix algebra unity is any
square matrix whose top left to
bottom right diagonal consists of
1s where all the rest of the matrixconsists of zeros
Matrices are only equal where
they are the same size and havethe same elements in the same
place, i.e.
I =1 0
0 1or I =
1 0 0
0 1 0
0 0 1
or I =
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
1 0 0
0 1 00 0 1
1 0
0 1
UNITY MATRIX
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Lesson 10 - 9
Year 1
CS113/0401/v1
UNITY MATRIX
As wit normal numbers where a number
multiplied by one equals itself (3 x 1 =
3) so with matrices, A matrix multiplied
by the unity matrix equals itself, i.e.
AI = A and IA = A
1 00 1
A =1 0
0 1for example
AI = x =1 62 3 1x1+6x0 0x6+0x30x1+1x2 0x6+1x3
=1 6
2 3
Similarly1 6
2 3IA =
1 0
0 1x 1x1+6x0 1x0+6x1
2x1+3x0 2x0+3x1=
= 1 6
2 3 thus proving that Al = IA = A
Note: The unit matrix, I, must always be square.
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Lesson 10 - 10
Year 1
CS113/0401/v1
EQUIVALENT MATRIX
Two matrices are equal if and only if
their corresponding elements are
equal. For instance, if
then matrix A = matrix B
Example:
a. Find the values of x and y if A + B = C
b. Is BC + CB?
c. Evaluate 3B
A = 2 34 5
2 3
4 5and B =
Given A = x 21 y
B =, 3 -54 2
8 -3
5 0and C =
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Lesson 10 - 11
Year 1
CS113/0401/v1
a. A +B = x 21 y
+ 3 -54 2
X+3 2-5
1+4 y+2=
= X+3 -35 y+2
X+3 -3
5 y+2= 8 -3
5 0
X-3 = 8 and y + 2 = 0
EQUIVALENT MATRIX
Solution:
Since A + B = C
Therefore x = 5, y = 2
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Lesson 10 - 12
Year 1
CS113/0401/v1
b. BC =3 -5
4 2
8 -3
5 0
= 24-25 -9+0
32+10-12+0= -1 -9
42 -12
CB =3 -5
4 2
8 -3
5 0
=24-12 -40-6
15+0 -25+0
=12 -46
15 -25
c. 3B = 33 -5
4 2
= 3x3 3(-5)3x4 3x2
= 9-1512 6
EQUIVALENT MATRIX Solution:
Thus BC = CB
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Lesson 10 - 13
Year 1
CS113/0401/v1
A group operates a chain of filling
stations in each of which are employed
cashiers, attendants and mechanics as
shown
The number of filling stations are
How many of the various types of staff are
employed in Southern England and inNorthern England?
Large stations
Medium stations
Small stations
Southern England
3
5
12
Northern England
7
8
4
Matrix B, i.e. 3 x 2
Types of filling station
Cashier
Attendants
Mechanics
Large
4
12
6
Medium
2
6
4
Small
1
3
2
Matrix A, i.e. 3 x 3
Exercise
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Lesson 10 - 14
Year 1
CS113/0401/v1
Solution
A is a 3 x 3 matrix, B is a 3 x 2 matrix
therefore AB is feasible and will be a 3 x
2 matrix.
A x B = AB
4 2 1
12 6 3
6 4 2
3 7
5 8
12 4
X11 X12
X21 X22
X31 X32
X11 = (4x3) + (2x5) + (1x12) = 34
X12 = (4x7) + (2x8) + (1x4) = 48
X21 = (12x3) + (6x5) + (3x12) = 102
X22 = (12x7) + (4x5) + (3x4) = 144
X31 = (6x3) + (4x5) + (2x12) = 62X32 = (6x7) + (4x8) + (2x4) = 82
Cashiers
AttendantsMechanics
South
3
512
North
7
84
AB is
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Lesson 10 - 15
Year 1
CS113/0401/v1
1
2
1 2 3 4 5
3
4
5
6
6
C
C
A B
A B
X
y
TRANSFORMATIONA transformation is an operation which
transform a point or a figure into another
point or figure.
TranslationA translation is a transformation which
moves all points in a place through the
same direction.
e.g. The triangle ABC has been transformed
onto the triangle ABC by a translation [ ]i.e. 3 squares to the right and 2 squares up in
the plane of the paper.
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Lesson 10 - 16
Year 1
CS113/0401/v1
Point a is mapped onto A by a
translation , denoted by T.
Enlargement (E)
An enlargement with centre O, scale
factor k is a transformation which
enlarges a given figure by k times the
original size.
If k > O, the given figure and its image
are on the same side of the centre of
enlargement O.
If k > O, the given figure and its image
are on opposite sides of O.
3
2
X
y
1
1
+
+
T =X
y
3
2 =4
3
Translation
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Lesson 10 - 17
Year 1
CS113/0401/v1
OA OB OC= = =
OA OB OC
o
C
C
A
A
BB
Under an enlargement,Area of image
= k2Area of Figure
Enlargement (E)The figure and its image after an
enlargement are similar, The scale factor K
If the image of a point (x,y) under a
transformation is the point itself i.e. (x,y),the point (x,y) is called an invariant point of
the transformation.
If a line is mapped onto itself under a
transformation, the line is said to be an
invariant linr under the transformation.
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Lesson 10 - 18
Year 1
CS113/0401/v1
A reflection is a transformation which
reflects all points of a plane in a line( on the plane ) called the mirror line.
ABC is mapped onto ABC under a
reflection in the line XY which is the
perpendicular bisector of AA, BB OR CC.
Under a reflection, the figure and its image
are congruent.
Example:
x
AA
CC
BB
Y
Reflection
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Lesson 10 - 19
Year 1
CS113/0401/v1
Rotation (R)
A rotation is a transformation which
rotates all points on plane about a fixed
point known as the centre of rotation,
6through a given angle in anti-clockwise
of clockwise direction.
The angle through which the points are
rotated is called the angle of rotation.
The triangle ABC is rotated about the
origin O through 90 in the anti-clockwise
direction, and mapped onto triangle ABC.
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Lesson 10 - 20
Year 1
CS113/0401/v1
OBC is mapped onto OBC under a shear along
the x-axis with factor k.
OC 6
K = = = 2OC 3
Shearing (H)A shear parallel to the x-axis is a
transformation which moves a point (x,y)
parallel to the x-axis through a distance
ky, where k is the shear factor.
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Lesson 10 - 21
Year 1
CS113/0401/v1
difference in y-coordinates of corresponding pointsk =
x-coordinates of original point
Shearing (H)
A shear parallel to the y-axis is a
transformation which moves a point (x,y)
parallel to the y-axis through a distance kx,
where k is the shear factor.
Stretching (S)
One way stretch
A stretch parallel to the x-axis is a
transformation which move a point
(x,y) parallel to the x-axis, through adistance ky, where k is the stretch
factor.
A stretch parallel to the y-axis is a
transformation which moves a point
(x,y) parallel to the y-axis through a
distance ky, where k is the stretchfactor.
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Lesson 10 - 22
Year 1
CS113/0401/v1
distance of new point the invariant linek =
x-coordinates of original point
Shearing (S)
In the case of stretching parallel to the x-
axis, the invariant line is the x-axis.
In the case of stretching parallel to the y-
axis, the invariant line is the y-axis.
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Lesson 10 - 23
Year 1
CS113/0401/v1
Shearing (S)
Two Way Stretch
If a figure is stretched parallel to the x-
axis as well as parallel to the y-axis, then
the stretch is called a two-way stretch.
Under a two-way stretch with h and k as
constants of stretch parallel to the x-axis
and y-axis respectively a point (x,y) is
mapped onto (hx,ky).
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Lesson 10 - 24
Year 1
CS113/0401/v1
Example: Matrix represents a
transformation T.Given (x,y) is the image of the point (a,b)
under the transformation T, find x and y in
terms of a and b.
Solution: Write the ordered pairs, (a,b)
and (x,y) as column vectors:
Premultiply by the matrix ,
we get
Therefore, x = a + 3b, y = 2a - 5b
1 3
2 -5
a
band
x
y
a
b
1 3
3 -5
x
y =1 3
2 -5
ab
=1xa + 3xb
2xa + (-5)xb
a + 3b
2a -5b=
Shearing (S)
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Lesson 10 - 25
Year 1
CS113/0401/v1
The matrix defines a
transformation T which maps the points
(a,b) onto ( a + 3b, 2a - 5b ).
Example:Find the coordinates of the
image of the point (-3,2) under
the transformation represented
by the matrix
Solution:Let the image of the point = (x,y)
Therefore the images of the point = (-11,-15)
1 3
2 -5
3 -15 0
X
y=
3 -1
5 0
-3
2
3x(-3) + (-1)x25x(-3) + 0x2=
=
=-9-2
-15+0
-11
-15
Stretching (S)
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Year 1
CS113/0401/ 1
Example:Find the matrix of the
transformation which maps
(1,0 ) onto (4,1) and (0,1)
onto (3,2).
Solution:Let the matrix of transformation
=
(1,0) (4,1)
because (4,1) is the image of (1,0)
1
0
Stretching (S)
a b
c d
4
1
a bc d=
4
1=
a +0c +0
=ac
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