Multiplying a Matrix to another Matrix

when multiplying two matrices, two sizes must be m x n and n x p and the resulting size is m x p.

2 x 3 ( m x n) and 3 x 2 ( n x p ) results to a 2 x 2 ( m x p) matrix.

you just multiply the 1st row to its corresponding column member and get the sum. then you get your first entry which in this case is 58.

 

(1 2 3) x (7 9 11) = (1 x 7) + (2 x 9) + (3 x 11) = 58

then just do the same thing again but do it with the first row then the second column.

(1 2 3) x (8 10 12) = (1 x 8) + (2 x 10) + (3 x 12) = 64

then just repeat it with the second row and 1st column then second row and 2nd column and you get the final matrix:

FIN!

Scalar Multiplication ...

Scalar Multiplication states that:

If matrix A = [aij] and c is a scalar, then cA = [caij].

you can use a combination of addition and scalar. for example
 solving a matrix m x n  A-B.

 A-B is just the same as A+(-B). you just do this by multiplying -1 to matrix B.

you can just do this if both matrix is the size m x n

Operations with Matrices

Definition of Equality of matrices:
Two matrices are said to be equal if they have the same size m x n .

Matrix Addition:
If A= [aij] and B= [bij] are matrices of the size m x n then

A+B= [aij + bij]

The sum of matrices with two different sizes are considered undefined.

Polynomial Curve Fitting....

Polynomial Curve fitting is a procedure that is done to know one polynomial function p(x) passes through given points.

If you have a collection of data represented by n point on an xy-plane.

(X1, Y1) (X2, Y2) ... (Xn, Yn)

then the polynomial function will have a degree of n-1 with the resulting p(x):

 p(x) = a₀ + a₁x + a₂x² + . . .  + a_n-1x^n-1

^{To solve the n coeffiecients of p(x), substitute each of the n points into the polynomial function and obtain n linear equations in n  variables.}

Network Analysis....

Network analysis are made up of one or more Junctions that are inter-connected in some way.It states that the amount that flows in is equal to the amount that flows out. These networks can be represented into a system of linear equations.

this is an example of a network with one junction. it can be expressed into the equation X1 + X2 = 25
(note the direction of the arrows).

These networks can be used into representing traffic problems, economic problems and with Kirchhoff's Law

Kirchhoff's Law is used when representing Circuits by the use of networks. It states that:

1.The sum of the current that goes in a junction is equal to the amount of current that comes out of it.

2.In a Closed path,the sum of the product of the resistance (ohms) and Current (Amperes) is equal to total voltage (Volts) of the path .

* the direction of the circuit sometimes depend on the battery symbol. the longer side is the output and the shorter side is the input.


example of a network of circuits that applies to Kirchhoff's Law. (note the direction of the circuit).

Row-Echelon in matrices

a matrix is said to be in row echelon form if:

  a.) all of the leading coefficients is 1.
  b.) each of the succeeding row from the bottom has the leading coefficient of 1
  c.) if the bottom row has all zeros (consistent dependent system).

an example of a matrix in row-echelon form.

                                                   
a matrix is said to be in reduced row-echelon form if:
 a.) it is in row-echelon form.
 b.) every leading coefficient is 1 and is the only non zero entry. (besides the constant).

an example of a matrix in reduced  row-echelon form.

Gaussian Elimination...

-also known as Row Reduction 

-an algorithm for solving systems of linear equations. It is usually understood as a sequence of operations performed on the associated matrix of coefficients. This method can also be used to find the rank of a matrix, to calculate the determinant of a matrix, and to calculate the inverse of an invertible square matrix. The method is named after Carl Friedrich Gauss, although it was known to Chinese mathematicians as early as 179 AD.


                            Here is an example of a augmented matrix to a row-echelon form.

Matrices...

Matrices are one of the important elements in Linear Algebra.

A Matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. The individual items in a matrix are called its elements or entries.

This is the notation of a matrix.

There are different row operations when it comes to solving a solution using the matrices.

There are three types of row operations:
1.row addition, that is adding a row to another.
2.row multiplication, that is multiplying all entries of a row by a non-zero constant;
3.row switching, that is interchanging two rows of a matrix;

Introduction to Linear Algebra

Linear Algebra is the study of linear sets of equations and their transformation properties. Linear algebra allows the analysis of rotations in space, least squares fitting, solution of coupled differential equations, determination of a circle passing through three given points, as well as many other problems in mathematics, physics, and engineering. Confusingly, linear algebra is not actually an algebra in the technical sense of the word "algebra" (i.e., a vector space over a field , and so on).

The matrix and determinant are extremely useful tools of linear algebra. One central problem of linear algebra is the solution of the matrix equation




for . While this can, in theory, be solved using a matrix inverse




other techniques such as Gaussian elimination are numerically more robust.

In addition to being used to describe the study of linear sets of equations, the term "linear algebra" is also used to describe a particular type of algebra. In particular, a linear algebra over a field has the structure of a ring with all the usual axioms for an inner addition and an inner multiplication together with distributive laws, therefore giving it more structure than a ring. A linear algebra also admits an outer operation of multiplication by scalars (that are elements of the underlying field ). For example, the set of all linear transformations from a vector space to itself over a field forms a linear algebra over . Another example of a linear algebra is the set of all real square matrices over the field of the real numbers.

First Post :)

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