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
matrix in Example 1 has three rows and two
columns, so
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 the unknowns,
is the constant of the equation.
TYPES O
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 such that
a b 3
A 1
3 c d
and
4 5 3e
B
2 6 7
This
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 is in (i.e., is
closed under the operation ).
a) , for
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 there is any, appear at the bottom of the matrix
The first
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 is to describe how a
function behaves as the independen
#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;
double a[n][n+1],x[n],l,d;
cout<"\n Enter the Elements o
#include <iostream>
using namespace std;
class Gauss
cfw_
float a[50][50];
int n;
public:
void VALUES()
cfw_
cout<"\t GAUSSIAN and GAUSS-JORDAN ELIMINATION PROGRAM\n\t\t\tMATH15-1
PROJ\n\n";
cout<"\nSubmitted by:\nCristobal, John Carlo D. \nDomingo, Adria
#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;
double a[n][n+1],x[n],l,d;
cout<"\n Enter the Elements o
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):
Presentation (/20):
TOTAL
Engr. Ericson D. Dimaunahan
Ins
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 that has its tail at one point and
the head of the arrow a
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
Has no solution
b.
INCONSISTENT
Has infinitely many sol
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 zeroes), if
it exists, appear at the bottom of the mat
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 content and state-of-the-art in delivery; by engaging in
cut
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 smaller. We
then move to the second number and do the sa
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 a basis
Find a basis that includes the given set of vec
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 properties of eigenvalues and eigenvectors
Determine the eigenva
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 subspaces.
Express a given vector as linear combination
of
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 is linear
transformation.
Perform the transformation o
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 +5y =11
C. x -6xy =
-20
5x y = 9
x 2y
-3y= -4
B. 2x + y