fundamentals of engineering analysis egr 1302 unit 1, lecture a
DESCRIPTION
Fundamentals of Engineering Analysis EGR 1302 Unit 1, Lecture A Approximate Running Time - 12 minutes Distance Learning / Online Instructional Presentation Presented by Department of Mechanical Engineering Baylor University Procedures: - PowerPoint PPT PresentationTRANSCRIPT
Slide 1 © 2005 Baylor University
Fundamentals of Engineering AnalysisEGR 1302 Unit 1, Lecture A
Approximate Running Time - 12 minutesDistance Learning / Online Instructional Presentation
Presented byDepartment of Mechanical Engineering
Baylor University
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Slide 2 © 2005 Baylor University
y=mx+b
x
y
x
y
z
y – mx = b
rearranged to be
ax + by = d
OR: ax +by+cz = d
Linear Systems
a1x1 + b1x2 + c1x3 + d1x4 = e1
a2x1 + b2x2 + c2x3 + d2x4 = e2
a3x1 + b3x2 + c3x3 + d3x4 = e3
a4x1 + b4x2 + c4x3 + d4x4 = e4
Slide 3 © 2005 Baylor University
How Do We Manage Large Amounts of Data?
Matrix Algebra
We arrange data in a:
Matrix = Table = Array
The key is learning the
Definitions
Symbology
Notation
Slide 4 © 2005 Baylor University
Basics of Matrix Notation
Denoted by Capital Letters A, B, C …
1 2 3 4 5 67 8 9 5 4 39 8 7 6 5 43 5 7 2 4 6
A =
A Matrix is referred to by
Row first, then column.
Row - Column
m = # rows n = # columns A is an “m by n” or “m x n”matrix
This matrix A is a 4x6 4x6 is the “Order” of the matrix
Slide 5 © 2005 Baylor University
B =
1 2 3 4 5 67 8 9 5 4 39 8 7 6 5 43 5 7 2 4 6
A =
1 2 3 4 5 53 4 1 7 8 35 6 0 2 1 79 8 7 6 5 40 3 5 7 4 2
Elements of a Matrix
Each element is denoted by lower case
aij i row, j column
a11 = 1
b34 = 6
Slide 6 © 2005 Baylor University
Order of Matrices
A= [3]1x1 a scalar B= [1 2 3 4] a row matrix C=
1234
a column matrix
A row matrix is a “1 X n” A column matrix is a “m X 1”
B= [1 2 3 4] is a “1 X 4” row matrix
Row or column matrices are also referred to a “Vectors”
A vector has magnitude and direction: [x,y,z]
The coordinates of a vector are represented with a matrix
Slide 7 © 2005 Baylor University
The Square Matrix
All matrices are “rectangular”, but … When “m = n”, the matrix is
“Square” or “n X n”
A = 3 1 42 0 56 4 2
“A” is a “3 X 3” square matrixx
a21 = 2 a12 = 1
Slide 8 © 2005 Baylor University
Basic Rules of Matrices
1. Equality – two matrices are equal if
- They are both the same “order”
- All respective elements are equal
In other words
aij = bij
A = a bc d
B = 2 x4 z
a = 2b = xc = 4d = z
When A = B
Slide 9 © 2005 Baylor University
Basic Rules of Matrices (cont.)
2. Multiply a matrix by a constant
Given k*A, where k = 2, and A = -3 2 1 4
2A = -6 4 2 8
Factoring: if C = 5 1015 20
1 23 4 Also C = 5 * Is not “C”!
Slide 10 © 2005 Baylor University
Basic Rules of Matrices (cont.)
3. The Null Matrix
- All elements are Zero
A = 0 00 0
A is a Null Matrix
Slide 11 © 2005 Baylor University
Basic Rules of Matrices (cont.)
4. Adding and Subtracting Matrices
- Must be of the same Order
A + B = C, only if
A is a “m x n” and B is a “m x n” then C is a “m x n”
aij + bij = cij
A = 2 -13 6
B = 0 32 -1
A+B = C = 2 25 5
Subtraction: (A – B) is the same as A+ (-1)*B
Slide 12 © 2005 Baylor University
Basic Rules of Matrices (cont.)
5. Associative Law
(A + B) + C = A + (B + C)
k*(A + B) = k*A + k*B
Now for a review of this lesson -
Slide 13 © 2005 Baylor University
Review of Matrix Rules
- Table or Array
- Capital Letters – “A”
- Rectangular or Square
- Order: m x n, or m=n is square ( n x n)
- m = #rows, n = #columns – always “row-column”
A + B Must be same Order
A = B if all respective elements are equal, and same Order
Element denoted by lower case aij
A = a11 a12
a21 a22
Slide 14 © 2005 Baylor University
This concludes Unit 1, Lecture A