MATRICES
MATH 15 - Linear Algebra
MATRICES
MATRICES
The size of a matrix is described in terms of the
number of rows (horizontal lines) and columns
(vertical lines) it contains. For example, the first
VECTOR SPACES
MATH 15 - Linear Algebra
DEFINITION 1
A real vector space is a set of
elements together with
two operations and satisfying the following properties:
() If and are any elements of , then
MODULE 1.3
MATRIX IN REDUCED ROW ECHELON FORM (RREF) AND APPLICATIONS
A matrix is said to be in reduced row echelon form if it satisfies the following properties:
(a)
(b)
(c)
(d)
All zero rows, if the
LIMITS
of
FUNCTIONS
LIMITS OF FUNCTIONS
OBJECTIVES:
define limits;
illustrate limits and its theorems; and
evaluate limits applying the given theorems.
DEFINITION: Limits
The most basic use of limits
#include<iostream>
using namespace std;
int main()
cfw_
int i,j,k,n;
cout<" L I N E A R A L G E B R A P R O J E C T :"<endl;
cout<"<endl;
cout<"<endl;
cout<" Enter the size of the matrix : ";
cin>n;
d
#include<iostream>
using namespace std;
int main()
cfw_
int i,j,k,n;
cout<" L I N E A R A L G E B R A P R O J E C T :"<endl;
cout<"<endl;
cout<"<endl;
cout<" Enter the size of the matrix : ";
cin>n;
d
MATRICES
DEFINITION, TYPES,
OPERATIONS AND
ALGEBRAIC PROPERTIES
LEARNING OBJECTIVES
1. Define a matrix.
2. Determine the size of a given matrix.
3. Identify the row vectors and column vectors of a giv
MAPUA INSTITUTE OF TECHNOLOGY
Intramuros, Manila
Department of Mathematics and Mechanics
MATH 15-1 / Linear Algebra
Quiz No.3
Set B
4th Term SY 2015-2016
Perform the indicated operation.
1. Confirm by
A Simple Machine Program Presented to
Mr. Jose Elvena Jr.
Mapa Institute of Technology
In Partial Fulfillment in the Requirements for
Linear Algebra (Math 15-1)
By:
Angelo Von Adiao
Ralph Gerald D. Ag
MAPUA INSTITUTE OF TECHNOLOGY
Department of Physics
E101: RESOLUTION OF FORCES
LAUENGCO, Troy Joseph D.
2015111250 BSME-2 Group 1
PHY10L-A1
SCORE:
Group Report (/40):
Analysis and Conclusion (/40):
Pr
MODULE 1.2
MATRIX OPERATIONS AND PROPERTIES
Matrix Equality - two matrices are equal if they have the same order and their corresponding entries are equal.
Example. Let matrix A be equal to matrix B s
LINEAR SYSTEMS
MATH 15 - Linear Algebra
LINEAR SYSTEM
DEFINITION:
A set of two or more linear equations
With the same solution set
where
are the numerical coefficients,
are the literal coefficients or
VECTOR SPACES
MATH 15 - Linear Algebra
DEFINITION 1
DEFINITION 1
DEFINITION 1
DEFINITION 1
Ways of Representing Vectors
Ways of Representing Vectors
Ways of Representing Vectors
Ways of Representing V
LINEAR SYSTEMS
MATH 15 - Linear Algebra
LINEAR SYSTEM
TYPES OF LINEAR SYSTEM
DEPENDING ON THE
NUMBER OF SOLUTION
LINEAR SYSTEM CAN BE
Has a unique or single solution set
CATEGORIZED AS:
a. CONSISTENT
ROW-ECHELON FORM
AND REDUCED ROWECHELON FORM
MATH 15 - Linear Algebra
DEFINITION
A matrix is in Reduced Row Echelon Form (RREF)
if it satisfies the following:
A.
A zero row (row containing entirely of
MATRIX OPERATIONS
AND PROPERTIES
MATH 15 - Linear Algebra
DEFINITION
EQUALITY OF MATRICES
DEFINITION
ADDITION AND SUBTRACTION
DEFINITION
SCALAR MULTIPLES
DEFINITION
DEFINITION
MULTIPLYING MATRICES
MUL
MAPUA INSTITUTE OF TECHNOTOGY
Department of Mathematics
vlsloN
The Mapua lnstitute of Technology shall be a global center of excellence in education by
providing instructions that are current in conte
DETERMINANT
MATH 15 - Linear Algebra
PERMUTATION
PERMUTATION
INVERSION
INVERSION
Solution:
a)
(3, 1, 4, 2)
We will start at the left most number and count the
number of numbers to the right that are s
BASIS and DIMENSION
MATH 15 - Linear Algebra
Department of Mathematics
OBJECTIVES
At the end of this lesson, the
students are expected to :
Define a basis and dimension
Determine whether vectors form
EIGENVALUES AND
EIGENVECTORS
MATH 15 - Linear Algebra
Department of Mathematics
OBJECTIVES
At the end of this lesson, the
students are expected to :
Define eigenvalues and eigenvectors
Prove propertie
SUBSPACES
MATH 15 - Linear Algebra
Engr. Dan Andrew Magcuyao
Department of Mathematics
OBJECTIVES
At the end of this lesson, the
students are expected to:
Define a subspace
Identify the properties sub
LINEAR
TRANSFORMATION
MATH 15 - Linear Algebra
Department of Mathematics
OBJECTIVES
At the end of this lesson, the
students are expected to:
Define linear transformation.
Determine if a transformation
MATH 15 - Linear Algebra
VECTOR
A vector is a quantity that has both magnitude and
direction. Mathematically, we can represent a
vector graphically in the plane by a directed line
segment or arrow tha
LINEAR EQUATION
A
linear equation in the variables
x.,x
is an equation that can
n n benwritten
in the form ax + ax + . + a
x =b
SYSTEM OF EQUATION
is
a set of more than one equation.
Examples
A. 3x