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