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# Linear Algebra

Linear Algebra usually consists of the linear set of equations as well as their transformations on it. It includes various topics such as matrices, vectors, determinants etc. Matrices and determinants are two very important topics of the linear algebra.

Linear Equations includes the topics mentioned as follows:

• Linear Equations

• Matrices

• Determinants

• Complex numbers

• Second degree equations

• Eigen values / Eigen vectors

• Vectors and its related operations

 Related Calculators Linear Calculator Algebra Calculator Algebra Division Algebra Factoring

## Introduction to Linear Algebra

Linear algebra is the branch of mathematics concerning vector spaces, Matrices, vectors, transformations, eigenvectors/values, solving system of linear equations.

Let us find the inverse of A. If A = $\begin{bmatrix} 1&0 &-2 \\ 3& 1 &4 \\ 5& 2 & -3 \end{bmatrix}$

Given

|A| = -13 $\neq$ 0, so inverse exists.

Find the adjoint for A

a$_{11}$ = -3 - 8 = -11

a$_{12}$ = -(-9 - 20) = 29

a$_{13}$ = 6 - 5 = 1

a$_{21}$ = -(0 + 4) = -4

a$_{22}$ = -3 + 10 = 7

a$_{23}$ = -(2 - 0) = -2

a$_{31}$ = 0 + 2 = 2

a$_{32}$ = -(4 + 6) = -10

a$_{33}$ = 1 - 0 = 1

adj A = $\begin{bmatrix} -11&29 &1\\ -4& 7 &-2 \\ 2& -10 & 1 \end{bmatrix}$

A-1 = $\frac{1}{|A|}$ adj A

## Elementary Linear Algebra

Elementary linear algebra is concerning with the following topics: linear equations, matrices, determinant, complex numbers eigenvalues and eigenvectors, rank of matrices, transformation.

Let us find the eigen values of A. Where A = $\begin{bmatrix} 2&0 &0\\ 3& 5 &0 \\ 1& 2 & -3 \end{bmatrix}$

det of ($\lambda$ I - A) = $\begin{vmatrix} \lambda - 2&0 &0\\ -3& \lambda - 5 &0 \\ -1& -2 & \lambda + 3 \end{vmatrix}$

= ($\lambda_1$ - 2)($\lambda_2$ - 5)($\lambda_3$ + 3)

Therefore, the eigen values of A are 2, 5, -3.

## Advanced Linear Algebra

There are various branches of linear algebra as follows:

Linear Equations: It is an equation of first degree for example, 2x + 5 = 9. We have to solve it and find the value of unknown variable in it.

Matrices: They are usually expressed as an array of numbers. For example Determinants: It is a term which is calculated for each and every possible matrix. It defines the nature of the matrix.

Complex numbers: They are used to represent complex variables onto the space For eg : 2 + 3i where 2 is the real part and 3i is the complex part.

Second degree equations: They are often called as the quadratic equations. various methods are used to solve them like factoring, by making the square, the quadratic formula etc are present in linear algebra.

Eigen values: These are the scalars which are associated with the linear set of equations also known as the characteristic values and used with respect to matrices .

Vectors and its related operations: Vectors are the entities having both magnitude as well as direction. Different operations can be applied on it like addition, subtraction, multiplication etc.

## Applications of Linear Algebra

Linear algebra has a major number of applications. It is one of the very essential branch of mathematics. Some of its applications are as folloed:

• Constructing curves

• Least square approximation

• Traffic flow

• Electrical circuits

• Determinants

• Graph theory

• Cryptography

## Linear Algebra Practice Problems

A = $\begin{bmatrix} 8&3 &3\\ 3& 5 &1 \\ 1& 4 & -3 \end{bmatrix}$