Title : Review of Mathematical Foundations
Intended Learning Outcomes :
At the end of this lesson, you should be able to :
1. Perform Matrix Operations (Multiplication, Transpose, Determinant)
2. Define the physical characteristics of Derivatives and Integrals
3. Calculate numerical values using Taylor Series Expansion
Discussions :
1. Matrix Operations
a. Definition of a Matrix
A matrix is a collection of numbers arranged in rectangular array. A matrix has a size mxn
where m is the number of rows while n is the number of columns. The elements of the matrix is
enclosed in a bracket. Suppose we have a matrix A with elements shown below. Element in the
second row and third column a23 is equal to 6.
12 4
A= 3 8 6
51 7
[ ]
b. Multiplication of Matrices
Let us define matrices A and B and C = AB. Matrix A is known as a pre-multiplier while
matrix B is the post-multiplier.
[ ]
[ ]
123
B= 0 2 0
512
12 4
A= 3 8 6
51 7
To calculate the elements of matrix C, multiply Arow by Bcolumn and add the result.
c11 = 1(1) + 2(0) + 4(5) = 21
Matrix multiplication is not commutative, AB ≠ BA.
c. Transpose of a Matrix
To transpose a matrix is to interchange its rows and columns.
T
[ ][ ]
124
135
A T= 3 8 6 = 2 8 1
517
467
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d. Determinant of a Matrix
The determinant of a matrix can be interpreted as a function that has an input of a square
matrix and that has an output of a number. If the determinant is nonzero, then the matrix is
invertible.
([ ])
det a b =ad−bc
cd
2x2 Matrix
3x3 Matrix
([ ])
a bc
det d e f =aei+bfg+cdh−afh−bdi−ceg
gh i
One way to visualize this is to copy the first and
second columns to the side of the matrix and multiply
using arrows as shown. Upward is negative and downward is positive.
Example :
[ ]
12 4
A= 3 8 6
51 7
det(A) = 1(8)7 + 2(6)5 + 4(3)1 - 5(8)4 – 1(6)1 – 7(3)2 = -80
The two main uses of the determinant :
1. to determine whether or not a matrix has an inverse
2. to determining the characteristic polynomial of a matrix, related to its eigenvalues.
2. Physical characteristics of Derivatives and Integrals
The derivative can be interpreted geometrically as the slope of the function at given point “a”. It
gives the instantaneous slope a that particular point. This is useful in determining speed, velocity
and acceleration for accelerated objects.
The integral can be interpreted geometrically as a the area under a function between points a
and b. The integral can applied to determining areas, volumes, surface areas, and moments inertia.
One can also determine the value of pi using the integral.
1
π=4∫ √ 1−x dx
2
0
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3. Taylor Series
A Taylor Series of a function is the sum of infinite terms, where each term has increasing
exponents like x, x2, x3, etc.
Function
Taylor Series
Sigma Notation
ex
e x=1+x+
sin x
cos x
2
3
4
x x
x
+ + +...
2! 3! 4!
∞
e =∑
x
n=0
sin x=x−
x3 x 5 x 7 x9
+ − + +...
3! 5 ! 7 ! 9!
cos x=x−
x2 x4 x6 x8
+ − + +...
2 ! 4! 6 ! 8 !
xn
n!
∞
(−1) n x 2n+1
sin x=∑
n=0 (2n +1) !
∞
cos x=∑
n =0
(−1)n x2n
(2n )!
We can approximate the value of a function at specific points using Taylor series. When
using sin and cos functions use radians mode for your calculator for a more accurate result.
Sample Problem : Show the increasing accuracy of Taylor series to determine the value of e3.
Solution :
e3 ≈ 20.0855
Terms
Taylor Series
Sum
1
1
1.0000
2
1+3
4.0000
2
3
1 + 3 + 3 /2!
2
8.5000
3
4
1+ 3 + 3 /2! + 3 /3!
5
1+ 3 + 32/2! + 33/3! + 34/4!
2
6
3
13.0000
4
16.3750
5
1+ 3 + 3 /2! + 3 /3! + 3 /4! + 3 /5!
2
3
4
5
18.4000
6
7
1+ 3 + 3 /2! + 3 /3! + 3 /4! + 3 /5! + 3 /6!
8
1+ 3 + 32/2! + 33/3! + 34/4! + 35/5! + 36/6! + 37/7!
2
9
3
4
5
6
19.4125
7
19.8464
8
1+ 3 + 3 /2! + 3 /3! + 3 /4! + 3 /5! + 3 /6! + 3 /7! + 3 /8!
2
10
3
4
5
6
7
8
20.0092
9
1+ 3 + 3 /2! + 3 /3! + 3 /4! + 3 /5! + 3 /6! + 3 /7! + 3 /8! + 3 /9!
2
3
4
5
6
7
8
9
20.0634
10
11
1+ 3 + 3 /2! + 3 /3! + 3 /4! + 3 /5! + 3 /6! + 3 /7! + 3 /8! + 3 /9! + 3 /10!
12
1+3+ 32/2! + 33/3! + 34/4! + 35/5! + 36/6! + 37/7! + 38/8! + 39/9! + 310/10! + 311/11!
13
14
2
3
4
5
6
7
8
9
10
11
20.0797
20.0841
12
1+3+ 3 /2! + 3 /3! + 3 /4! + 3 /5! + 3 /6! + 3 /7! + 3 /8! + 3 /9! + 3 /10! + 3 /11! + 3 /12!
2
3
4
5
6
7
8
9
10
11
12
13
20.0852
1+3+ 3 /2!+3 /3! + 3 /4! + 3 /5! + 3 /6! + 3 /7! + 3 /8! + 3 /9! + 3 /10! + 3 /11! + 3 /12! +3 /13! 20.0855
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Self Assessment(Problem Set) :
Show your complete solution.
[ ]
223
A= 3 2 5
512
[ ]
−1 2 4
B= −3 8 6
−5 1 7
1. Perform the indicated operations
a. AxB
b. A+B
c. det(A)
d BT
2. Find the slope of the fuction y = 2x2 + 1 at x = 1.
3. Determine the area of the under a curve y = 2x2 + 1 bounded between x =0 and x =1.
4. Show the increasing accuracy of Taylor series to determine the value of sin (1).
References :
1. Applied Numerical Methods for Engineers and Scientist
Steven C. Chapra, McGraw Hill 2nd Edition 2008
2. https://www.mathsisfun.com/algebra/taylor-series.html
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