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Linear Algebra

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Linear algebra is the branch of mathematics concerning vector spaces and their transformation properties. Linear algebra can be applied to pure mathematics as well as applied mathematics.
It facilitates the solution of linear systems of differential equations and is a well developed theory. Linear equations are common in science and mathematics, and is defined to be a mathematical statement of equality. Determination of value of the variable which satisfies an equation is called solution of the equation or root of the equation. An equation in which highest power of the variable is 1 is called a linear equation. This is also called the equation of degree 1.

A simple equation in one unknown $x$ is in the form $ax$ + $b$ = 0.
Where $a$ and $b$ are constants and $a \neq$ 0.
A simple equation has only one root.
This page explains about linear equations, elimination method, cross multiplication method, and vectors. 

Linear Algebra Terms

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Given below are some of the most commonly used terms in linear algebra.
Linear Equation
General form of a linear equations in two unknowns x and y is $ax$ + $by$ + $c$ = 0 where a and b are non zero coefficients.
Two such equations a$_{1}$x + b$_{1}$y + c$_{1}$ = 0 and a$_{2}$x+ b$_{2}$x + c$_{2}$ = 0 from a pair of simultaneous equations in $x$ and $y$. A value for each unknown which satisfies simultaneously both the equations will give the roots of the equations.

Matrix
Matrix is an array of numbers and is usually represented by a capital letter. Entries of an element is shown in a lowercase letter with a subscript of row, column. Mathematical operations can be performed on matrices. For adding and subtracting two matrices, add and subtract the numbers in their matching positions respectively. Product of two matrices is a matrix that represents the composition of two linear transformations. A square matrix will have an inverse if and only if its determinant is zero. Eigen values and eigen vectors gives an insight of linear transformations.

Variable
A variable is a symbol that represents a quantity in a mathematical expression.

Slope intercept form
It is used to express equation of a line, m is the slope and b is the y-intercept.
A way to express the equation of a line and is used most frequently.

 Example : $y$ = 8$x$ + 6, Represents an equation of a line with slope 8 and y intercept of 6.

Linear Algebra Formulas

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Given below are some of the linear algebra formulas.

1) Position vector of any point $L(x, y)$:
$OL$ = $\begin{pmatrix}
x\\
y
\end{pmatrix}$

2) Magnitude of position vector:
$OL$ = $\sqrt{x^{2}+y^{2}}$ and direction = $tan\theta$ =$\frac{y}{x}$

3) Scalar product of two vectors:
Let $\vec{x}$ = ($\vec{x_{1}},\vec{x_{2}}), \vec{y}=(\vec{y_{1}},\vec{y_{2}}$) then the scalar product of $\vec{x}$ and $\vec{y}$ is

$\vec{x}$ . $\vec{y}$ = $|\vec{x}|.|\vec{y}|cos\theta$

$\theta$ being angle between them.

4) Cross product of two vectors:
Let $\vec{x}$ = ($\vec{x_{1}},\vec{x_{2}},\vec{x_{3}}), \vec{y}= (\vec{y_{1}},\vec{y_{2}},\vec{y_{3}}$)be two vectors then the cross product of $\vec{x} \times \vec{y}$ is

|$\vec{x} \times \vec{y}$| = |$\vec{x}||\vec{y}$|sin$\theta$

5) Angle between two vectors:

$cos\alpha$ = $\frac{x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3}}{\sqrt{x_{1}^{2}+x_{2}^{2}+x_{3}^{2}}.\sqrt{y_{1}^{2}+y_{2}^{2}+y_{3}^{2}}}$

6) Scalar Multiplication:

$k$ . $\vec{x}$ = (k$\vec{x_{1}}+k\vec{x_{2}}+k\vec{x_{3}}$)

Linear Algebra Properties

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Linear algebra is central to both pure and applied mathematics. The following result lists the basic algebraic properties of the vectors operations and matrix operations.

Properties of matrix arithmetic:
Assume that the size of the matrices, M, N and T, are such that the indicated operations can be performed, the below rules of matrix arithmetic are valid.
1) Commulative law for addition: $M$ + $N$ = $N$ + $M$

2) Associative law for addition: $M$ + $(N + T)$ = $(M + N)$ + $T$

3) Associative law for multiplication: $M (NT)$ = $(MN) T$

4) Left distribution law: $M (N + T)$ = $MN$ + $MT$

5) Right distribution law: $(N + T) M$ = $NM$ + $TM$

Methods to solve system of linear equations:

Elimination method
In this method two given linear equations are reduced to a linear equation in one unknown by eliminating one of the unknowns and then solving for the other unknown.

Cross Multiplication Method

Let two equations be
a$_{1}$x + b$_{1}$y+c $_{1}$ = 0
a$_{2}$x+ b $_{2}$y + c $_{2}$= 0
Arrange the coefficients of $x$, $y$ and constant.

Find the value of $x$ and $y$.

$x$ = $\frac{x}{b_{1}c_{2} - b _{2}c_{1}}$ = $\frac{y}{c_{1}a_{2} - c_{2}a_{1}}$ = $\frac{1}{a_{1}b_{2} - a_{2}b_{1}}$

Properties of Determinant:

1) Let $A$ be a $n \times n$ matrix.
det(A) = det(A')

2) If two rows (columns) of A are equal then det(A) = 0

3) If a row of A consists entirely of 0, then det(A) = 0

4) det(A + B) is not always equal to det(A) + det(B)

5) The value of a determinant is unaltered if its rows and columns are interchanged.

6) If any two rows (columns) of a determinant are interchanged, then the sign of the determinant gets changed.

7) If any two rows (columns) of a determinant are identical, then the value of the determinant is zero.

8) If every element of row (column) of a determinant is multiplied by a scalar k, then the value of the determinant is multiplied by k.

9) If a row (column) of a determinant is a multiple of another row (column) then the value of the determinant is zero.

10) If A and B are square matrices of the same order, then |AB| = |A| |B|.

Linear Algebra Examples

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Example 1: $\frac{4x}{3}$ - 1 = $\frac{14}{15}$ $x$ + $\frac{19}{5}$

Solution: By transposing the variables in one side and the constants in other side we have

$\frac{4x}{3}$ - $\frac{14x}{5}$ = $\frac{19}{5}$ + 1

$\frac{(20 - 42)x}{15}$ = $\frac{19+5}{5}$

$\frac{-22x}{15}$ = $\frac{24}{5}$

$x$ = $\frac{24*15}{5*-22}$

$x$ = 3.27

Example 2: Solve 2$x$ + 5$y$ = 9 and 3$x - y$ = 5

Solution:
2$x$ + 5$y$ = 9    -----1
 3$x$ - $y$ = 5      ----- 2

Multiply 2 equation by 5.
15$x$ - 5$y$ = 25

By adding equation 1 and equation 3, we get
17$x$ = 34       --------3
$x$ = 2.

Substituting this value of $x$ in equation 1.
5$y$ = 9 - 2$x$
We find 5$y$ = 9 - 4
5$y$ = 5
$y$ = 1

Example 3: Solve 3$x$ + 2$y$ + 17 = 0 and 5$x$ - 6$y$ - 9 = 0

Solution:
Formula for finding the values of $x$ and $y$.

$x$ = $\frac{x}{b_{1}c_{2} - b _{2}c_{1}}$ = $\frac{y}{c_{1}a_{2} - c_{2}a_{1}}$ = $\frac{1}{a_{1}b_{2} - a_{2}b_{1}}$

=> $\frac{x}{84}$ = $\frac{y}{112}$ = $\frac{1}{-28}$

or $\frac{x}{3}$ = $\frac{y}{4}$ = $\frac{1}{-1}$

Hence $x$ = -3 and $y$ = -4