LINEAR ALGEBRA
2nd YEAR COLLEGE - FIRST SEMESTER
PRELIMINARY TERM
A SYSTEM HAS THESE PROPERTIES
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It consists of several parts which interact and affect one
another.
It produces an effect or output as a result of some cause or
input.
SYSTEM OF LINEAR EQUATION
LINEAR EQUATION
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A linear equation in n variables x1,x2,x3,...,xn has the form
a1x1+a2x2+a3x3+...+anxn=b
The coefficients a1,a2,a3,..,an are real numbers, and the
constant term b is a real number. The number a1 is the
leading coefficient, and x1is the leading variable.
A solution to the linear equation is a sequence of n
numbers s1,s2,s3,..,sn which has the property that the linear
equation is satisfied when x1=s1,x2=s3,x3=s3...,xn=sn
substituted in the linear equation.
degree is one
the variable is NOT involve in any transcendental function
NO product of variables
GAUSSIAN AND GAUSS-JORDAN ELIMINATION
MATRIX
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is a rectangular array of numbers denoted by:
ELEMENTARY ROW OPERATIONS
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To solve linear equations
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Elimination Method
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Substitution Method
Interchange two rows
Multiply a row by a nonzero constant
Add a multiple of a row to another row
ROW-ECHELON FORM (REF)
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All rows consisting entirely of zeros occur at the bottom of
the matrix.
● For each row that does not consist entirely of zeros, the
first nonzero entry is 1 (called a leading 1).
● For two successive (nonzero) rows, the leading 1 in the
higher row is farther to the left than the leading 1 in the
lower row.
Take note :
A matrix in row-echelon form is in reduced
row-echelon form (RREF) if every column that has a leading 1
has zeros in every position above and below its leading 1.
STEPS IN GAUSSIAN AND GAUSS-JORDAN ELIMINATION
1.
PARAMETRIC REPRESENTATION
STEP 1
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let the free variables as dummy variables (q,r,s,t,...)
STEP 2
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substitute the dummy variables to the original equation
Write the augmented matrix of the system of linear
equations.
2. Use elementary row operations to rewrite the augmented
matrix in row-echelon form.
3. Write the system of linear equations corresponding to the
matrix in row-echelon form, and use back-substitution to
find the solution.
REMARK: In Gauss- Jordan elimination continues the process of
reduction until a reduced row-echelon form is obtained.
STEP 3
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1
solve for leading variable
PREPARED BY: CLARKY
LINEAR ALGEBRA
2nd YEAR COLLEGE - FIRST SEMESTER
TRANSPOSE OF MATRIX
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rows become column, column become rows
The transpose of an m-by-n matrix A is the n-by-m matrix
AT (also denoted Atr or tA) formed by turning rows into
columns and vice versa: (AT)ij = Aji.
INVERSE OF MATRIX
SPECIAL TYPES OF MATRICES
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SQUARE MATRIX Is a matrix where m= n and that the
numbers a11,a22,a33…ann form the main diagonal of A.
DIAGONAL MATRIX An n x n matrix A = [ aij ] is called a
diagonal matrix if aij = 0 for i =1= j.Thus, for a diagonal
matrix, the terms off the main diagonal are all zero.
SCALAR MATRIX A scalar matrix is a diagonal matrix
whose diagonal elements are equal.
IDENTITY MATRIX The scalar matrix In = [dij ], where dij; = I
and dij = 0 for i not equal to j , is called the n x n identity
matrix.
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Let A be a square matrix of size n x n. If there exists a
matrix A-1 such that AA-1=A-1A=In, where In is the n x n
identity matrix , then is the inverse of A.
PROPERTIES
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If A is singular , then it does not have an
inverse
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If A is non-singular, then it has one and only
one inverse
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If , for n x n matrices, , then and
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(A-1)-1=A
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(AB)-1=B-1A-1
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(Ak)-1=A-1A-1A-1...A−1=(A-1)k, where k is a
positive integer
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(cA)-1=(1/c)A-1,wherec≠0
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(AT)-1=(A-1)T
INVERSE OF 2X2 MATRIX
MATRIX OPERATIONS
ADDITION AND SUBTRACTION OF MATRICES
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Formula:
element +/- element
SCALAR MULTIPLICATION
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constant c multiply by matrix m x n
c(element)
MATRIX MULTIPLICATION
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if A is m x n matrix and B is s x t matrix where n=s, then it
is valid to multiply, the result is m x t
(3x2)(3x2) can not be multiply
(3x2)(2x4) can, the result is 3x4
row by column
INVERSE OF MXN MATRIX (ANY MATRIX)
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PREPARED BY: CLARKY
create an augmented matrix of the original matrix and an
identity matrix
make the original matrix to an RREF
the matrix formed from identity matrix is the inverse of the
original matrix
LINEAR ALGEBRA
2nd YEAR COLLEGE - FIRST SEMESTER
interchange row operations
interchange
multiply row by non-zero constant
reciprocal
add a multiple of a row to another
row
negate
SLE USING INVERSE OF A MATRIX
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Multiply the inverse coefficient matrix to the constant
matrix
x=A-1b
LU-FACTORIZATION
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If the n x n matrix can be written as the product of a lower
triangular matrix L and an upper triangular matrix U, then
A = LU is an LU-factorization of A
1.
2.
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4.
get the coefficient and constant matrix; A and b
get the upper and lower triangular matrix; U and L
solve for y; Ly=b
solve for x; Ux=y
MIDTERM
LU - FACTORIZATION
ELEMENTARY MATRIX
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An n x n matrix is called an elementary matrix if it can be
obtained from the identity matrix I by a single elementary
row operation.
SLE USING LU FACTORIZATION
FACTORING OF MATRIX
ELEMENTARY ROW OPERATIONS
3
TO DO
PREPARED BY: CLARKY
LINEAR ALGEBRA
2nd YEAR COLLEGE - FIRST SEMESTER
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FINDING DETERMINANTS USING EXPANSION BY
COFACTORS
FINDING DETERMINANTS USING ELEMENTARY ROW
OPERATIONS
DETERMINANT
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is a unique number associated with a square matrix
2X2 MATRIX
Note: To find the determinant of a triangular matrix or diagonal
matrix of any nxn matrix, just multiply the diagonal.
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Minors (Mij)
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Cofactors (Cij) = (-1)i+j(Mij)
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TO DO
interchange row operations
change sign
multiply row by non-zero constant
multiply the det by k
add a multiple of a row to another
row
no change
get the upper triangular of the matrix, multiply the diagonal
3X3 MATRIX
MINORS AND COFACTORS
ELEMENTARY ROW OPERATIONS
CRAMER’S RULE
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PREPARED BY: CLARKY
to solve linear equation
LINEAR ALGEBRA
2nd YEAR COLLEGE - FIRST SEMESTER
ADJOINT
VOLUME OF TETRAHEDRON
or
TWO-POINT FORM OF THE EQUATION OF THE LINE
INVERSE OF A MATRIX GIVEN ADJOINT
FINAL TERM
VECTORS IN THE PLANE
A vector in the plane is represented geometrically by a directed
line segment whose initial point is the origin and whose
terminal point is the point (x1, x2)
This vector is represented by the same ordered pair used to
represent its terminal point. That is, x =(x1,x2) The coordinates x1
and x2 are called the components of the vector x. Two vectors
in the plane u = (u1, u2)and v=(v1,v2) are equal if and only if u1=v1
and u2=v2.
AREA OF A TRIANGLE
LINEAR COMBINATION
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PREPARED BY: CLARKY
LINEAR ALGEBRA
2nd YEAR COLLEGE - FIRST SEMESTER
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Properties
○ d(u,v)≥0
○ d(u,v)=0 if and only if u=v
○ d(u,v)=d(v,u)
MAGNITUDE AND DISTANCE BETWEEN VECTORS
VECTORS in the plane can be characterized as directed line
segments having a certain length and direction.
N
LENGTH OF VECTOR IN R
DOT PRODUCT AND ORTHOGONAL
EXAMPLE:
ANGLES BETWEEN TWO VECTORS
UNIT VECTORS IN THE DIRECTION OF V
If v is a nonzero vector in Rn, then the vector u=v/∥v∥ has the
length 1 and the same direction as v. This vector u is called the
unit vector in the direction of v
Formula:
ORTHOGONAL VECTORS
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Two vectors u and v in Rn are orthogonal if u∙v=0
EIGENVECTORS AND EIGENVALUES
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DISTANCE BETWEEN TWO VECTOR IN R
The distance between the two vectors u and v in Rn is
d(u,v)=∥u−v∥
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The formula to find eigenvalues and eigenvectors is Ax = λx,
where A is a given vector, x is called an eigenvector and λ
(lambda) is an eigenvalue.
PREPARED BY: CLARKY
LINEAR ALGEBRA
2nd YEAR COLLEGE - FIRST SEMESTER
TAPOS NA TOH. MWAA
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PREPARED BY: CLARKY
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