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Wednesday, August 13, 2014

Matrix Multiplication

Matrix Multiplication is a binary operation that takes a pair of matrices, and produces another matrix. Numbers such as the real or complex numbers can be multiplied according to elementary arithmetic

If A is an n × m matrix and B is an m × p matrix,
\mathbf{A}=\begin{pmatrix}
 A_{11} & A_{12} & \cdots & A_{1m} \\
 A_{21} & A_{22} & \cdots & A_{2m} \\
\vdots & \vdots & \ddots & \vdots \\
 A_{n1} & A_{n2} & \cdots & A_{nm} \\
\end{pmatrix},\quad\mathbf{B}=\begin{pmatrix}
 B_{11} & B_{12} & \cdots & B_{1p} \\
 B_{21} & B_{22} & \cdots & B_{2p} \\
\vdots & \vdots & \ddots & \vdots \\
 B_{m1} & B_{m2} & \cdots & B_{mp} \\
\end{pmatrix}

the matrix product AB (denoted without multiplication signs or dots) is defined to be the n × p matrix

\mathbf{A}\mathbf{B} =\begin{pmatrix}
 \left(\mathbf{AB}\right)_{11} & \left(\mathbf{AB}\right)_{12} & \cdots & \left(\mathbf{AB}\right)_{1p} \\
 \left(\mathbf{AB}\right)_{21} & \left(\mathbf{AB}\right)_{22} & \cdots & \left(\mathbf{AB}\right)_{2p} \\
\vdots & \vdots & \ddots & \vdots \\
 \left(\mathbf{AB}\right)_{n1} & \left(\mathbf{AB}\right)_{n2} & \cdots & \left(\mathbf{AB}\right)_{np} \\
\end{pmatrix}

If the column and rows are in the same size it is defined.



EXAMPLE:

Matrix Subtraction

Matrix Subtraction- you can subtract two matrices (of the same size) by adding their corresponding entries.

EXAMPLES: 

1.

2.
Given the following matrices, find A – Band A – C, or explain why you can not.
      A = [[ -1 2 0 ][ 0 3 6 ]] B = [[ 0 -4 3 ][ 9 -4 -3 ]] C = [[ 0 -4 ][ 9 -4 ]]
    A and B are the same size, each being 2 × 3 matrices

      A - B = [[ -1 2 0 ][ 0 3 6 ]] - [[ 0 -4 3 ][ 9 -4 -3 ]] = [[-1-0 2+4 0-3 ][ 0-9 3+4 6+3]]
    A and C are not the same size, since A is 2 × 3 and C is 2 × 2. So this subtraction is not defined.
      A - B = [[-1 6 -3 ][-9 7 9]]
      A – C is not defined, because A and C are not the same size.

Matrix Addition

 Matrix Addition is the operation of adding two matrices by adding the corresponding entries together.

\begin{align}
\bold{A}+\bold{B} & = \begin{bmatrix}
 a_{11} & a_{12} & \cdots & a_{1n} \\
 a_{21} & a_{22} & \cdots & a_{2n} \\
 \vdots & \vdots & \ddots & \vdots \\
 a_{m1} & a_{m2} & \cdots & a_{mn} \\
\end{bmatrix} + 

\begin{bmatrix}
 b_{11} & b_{12} & \cdots & b_{1n} \\
 b_{21} & b_{22} & \cdots & b_{2n} \\
 \vdots & \vdots & \ddots & \vdots \\
 b_{m1} & b_{m2} & \cdots & b_{mn} \\
\end{bmatrix} \\
& = \begin{bmatrix}
 a_{11} + b_{11} & a_{12} + b_{12} & \cdots & a_{1n} + b_{1n} \\
 a_{21} + b_{21} & a_{22} + b_{22} & \cdots & a_{2n} + b_{2n} \\
 \vdots & \vdots & \ddots & \vdots \\
 a_{m1} + b_{m1} & a_{m2} + b_{m2} & \cdots & a_{mn} + b_{mn} \\
\end{bmatrix} \\

\end{align}\,\!










EXAMPLE: 
[[ 0 1 2 ][ 9 8 7 ]] + [[ 6 5 4 ][ 3 4 5 ]] = [[ 6 6 6 ][ 12 12 12 ]]


Equality of Matrices

Two matrices are equal if all three of the following conditions are met:
  •  Each matrix has the same number of rows.
  •  Each matrix has the same number of columns.
  •  Corresponding elements within each matrix are equal.

Consider the three matrices shown below.



If A = B then we know that x = 34 and y = 54, since corresponding elements of equal matrices are also equal.
We know that matrix 
C is not equal to A or B, because C has more columns.

  •  Two equal matrices are exactly the same.
  •  If rows are changed into columns and columns into rows, we get a transpose matrix. If the original matrix is A, its transpose is usually denoted by A' or At.
  •  If two matrices are of the same order (no condition on elements) they are said to be comparable.
  • If the given matrix A is of the order m x n, then its transpose will be of the order n x m. 
  1. Column matrix - a matrix that has only one column.
  2. Row matrix - a matrix that has only one row.

Analysis of an Electrical Network

EXAMPLE: 

  • Applying Kirchhoff's Law to have the Linear equations in the variables.
  • Below the linear equations is the Augmented and Gauss-Jordan elimination.






Kirchoff's Law

Kirchhoff's Law is an analysis of such a system uses two properties of electrical networks or two equalities that deal with the current and potential difference in the lumped element model of electrical circuits.

1. All current flowing into a junction must flow out of it.
2. The sum of the products IR ( I is current and  R is resistance ) around a closed path is equal to the total voltage in the path.

EXAMPLE:
a.
b.
c.
d.
Solution: 






Network Analysis

Network Analysis - composed of branches and junctions are used as models in many diverse fields such as economics, traffic analysis, and electrical engineering.

EXAMPLE: 









Monday, July 21, 2014

Monday, July 7, 2014

Homogeneous System

A homogeneous system of linear algebraic equations is one in which all the numbers on the right hand side are equal to 0:

a11x1 + : : : + a1nxn = 0
...

am1x1 + : : : + amnxn = 0

EXAMPLE:
 


(0,0,0) - trivial or obvious solution



Gauss- Jordan Elimination

  • Gauss- Jordan Elimination
                    - A method of solving a linear system of equations. This is done by transforming the system's augmented matrix into reduced row-echelon form by means of row operations.

1. Write the Augmented Matrix
2. Target: Reduced Row Echelon Form
3. Rewrite in linear systems

Gaussian Elemination with Back Substitution

1. Write the augmented matrix
2. Use Elementary Row Operations to obtain Row Echelon form of the matrix
3. Rewrite in linear system
4.Back Substitution

Tuesday, June 17, 2014

Row-Echelon Form

 
In linear algebra, a matrix is in echelon form if it has the shape resulting of a Gaussian elimination. Row echelon form means that Gaussian elimination has operated on the rows and column echelon form means that Gaussian elimination has operated on the columns. In other words, a matrix is in column echelon form if its transpose is in row echelon form. Therefore only row echelon forms are considered in the remainder of this article. The similar properties of column echelon form are easily deduced by transposing all the matrices.


 
Specifically, a matrix is in row echelon form if
 
Example:

 A 3×5 matrix in row echelon form:


\left[ \begin{array}{ccccc}
1 & a_0 & a_1 & a_2 & a_3 \\
0 & 0 & 2 & a_4 & a_5 \\
0 & 0 & 0 & 1 & a_6
\end{array} \right]

Elementary Row Operations

Three operations the can be used on a system of linear equations to produce equivalent systems.
  1. Interchange two equations.
  2. Multiply an equation by a nonzero constant.
  3. Add a multiple of an equation to another equation.

Two types of Matrix

  • Augmented Matrix - the matrix derived from the coefficients and constant terms of a system of linear equations.
Example:

  • Coefficient Matrix - the matrix containing only the coefficients of the system.
Example:

Monday, June 16, 2014

Matrix





In mathematics, 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.
An example of a matrix with 2 rows and 3 columns is
\begin{bmatrix}1 & 9 & -13 \\20 & 5 & -6 \end{bmatrix}.
 
 
 
 
 
 
 

Saturday, June 14, 2014

What is 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. The solving of the simple linear equation ax + b = 0 may be
considered as the original problem of this subject.



History
The study of linear algebra first emerged from the study of determinants, which were used to solve systems of linear equations. Determinants were used by Leibniz in 1693, and subsequently, Gabriel Cramer devised Cramer's Rule for solving linear systems in 1750. Later, Gauss further developed the theory of solving linear systems by using Gaussian elimination, which was initially listed as an advancement in geodesy.

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