If A is an n × m matrix and B is an m × p matrix,
the matrix product AB (denoted without multiplication signs or dots) is defined to be the n × p matrix
If the column and rows are in the same size it is defined.
EXAMPLE:
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,
the matrix product AB (denoted without multiplication signs or dots) is defined to be the n × p matrix
If the column and rows are in the same size it is defined.
EXAMPLE:
If A is an n × m matrix and B is an m × p matrix,
the matrix product AB (denoted without multiplication signs or dots) is defined to be the n × p matrix
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.
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 ]]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_tlBH3AJpqs2f-8mx8vvV2lkRaiCxYqFnC9GQ0EOCWwtV8Li16T2ZcdaZxPf3S9vxpP0tnK9rxkkNHViICtJj4yAYUf_6h7b1uztK9lD4dcqMV7XpkU-RE=s0-d){kind=small}
![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]]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_t97_JTMUZjpRHGuxr2Iy6PZK3Y8WenWV2eeWwgUkfAM5-VrfEET_cIUsWdu5hK5vDpwi7JXJKMIimCtgCjGPQ9FRje48d5FMCg2qlNELujwXpKKmkK1A=s0-d)
- 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]]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vhyQtJwz_ektg8O5hftIdjhlKYA98pFYQjVL9rbTOKWMvde-c81WgOMoVflVHCep7GrxumEpW9Fc1td9XqSwZMl9jcyzyFgwDBf3AICRIYJyeS1B-esZ4=s0-d){kind=small}
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.
EXAMPLE:
![[[ 0 1 2 ][ 9 8 7 ]] + [[ 6 5 4 ][ 3 4 5 ]] = [[ 6 6 6 ][ 12 12 12 ]]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_s5dL4WW1gtX-5G5GWjKZbPdz8dvbNxKIM3s94VD896RhaR2zYjLD_XEJ20Z00DLPxymR6s4gm9Lbfnf3GNBO_bXKXtQayEE0ryJAuWaHPokn_gYyj5ow=s0-d)
EXAMPLE:
Equality of Matrices
Two matrices are equal if all three of the following conditions are met:
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.
- 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.
- Column matrix - a matrix that has only one column.
- 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:
EXAMPLE:
(0,0,0) - trivial or obvious solution
Gauss- Jordan Elimination
- Gauss- Jordan Elimination
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
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.
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]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_sVAeKa3x59XvUa_agvP1AsJGtNeuA5ZHesA4VyNYno2Pr0Qwvs5OjMcJtE9SYlQBOTAoP1OOsTKmPEBotzuvGtCZeE5qjVbElZOqdIYQgKWyxYj00-mCGdksY8qdM6KRUjZ3xOMhw-YQvj4npHaA=s0-d)
Elementary Row Operations
Three operations the can be used on a system of linear equations to produce equivalent systems.
- Interchange two equations.
- Multiply an equation by a nonzero constant.
- 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.
- Coefficient Matrix - the matrix containing only the coefficients of the system.
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
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.
Welcome to my Blog!
Hello! This is Ara, 15 years of age and I'm studying at Regional Science High School III. This blog will have most our lessons or topics discussed in our school.
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