# Linear Algebra Index

Linear algebra is the study of systems of equations and functions that map between vector spaces

## Notation

Vector | Notation

A variable that represents a vector often has an arrow over the top to indicate that it is a vector.

Magnitude | Notation

To vertical bars on either side of a vector represent taking the magnitude or length of that vector

Cross Product | Notation

In linear algebra the cross product operator is the times operator that is used in elementary mathematics.

Matrix | Notation

In Linear Algebra, a m by n matrix is denoted as a grid of numbers with two brackets on either side. The variable m corresponds to the number of rows, and the variable n corresponds to the number of columns.

Determinant | Notation

The syntax for a determinant is to vertical bars on either side of the matrix.

Scalar Product | Notation

The syntax for a scalar product is to angle brackets on either side of two vectors separated by a comma.

## Formulas

Angle Between Two Vectors | Formula

The angle between two vectors can be calculated using the arc-cosine of their dot product divided by the product of their magnitudes.

Area Between Two Vectors | Formula

The area between two vectors is given by the magnitude of their cross product. This formula is a generic way to find the area of any triangle given three points.

Cross Product | Formula

The cross product of two vectors can be calculated using the formal determinant.

Determinant of Three by Three Matrix | Formula

The formula for the determinant of a three by three matrix.

Determinant of Two by Two Matrix | Formula

The formula for the determinant of a two by two matrix.

Dot Product | Formula

The dot prodcut of two vectors is calculated by summing together the product of corresponding elements.

Dot Product Geometric | Formula

The dot product can be geometrically interpretted as the magnitude of the two vectors multiplied by the cosine of the angle between them.

Magnitude of Cross Product | Formula

The magnitude of the cross product can be given as the magnitude of the two vectors multiplied by the sine of the angles between them.

Magnitude of Vector | Formula

The magnitude of a vector is given by the square root of the sum of its components squared.

## Operators

Cross Product | Operator

The cross product operates on two vectors and produces another vector as a result.

Matrix Multiplication | Operator

Matrix multiplication composes two matrices together to create a third matrix function.

Matrix Transpose | Operator

The transpose of a matrix is an operator which reflects a matrix accross its diagonal.

## Concepts

Determinant | Concept

The determinant is a function that maps the values in a matrix to a number. Using this number, certain properties can be detirmined. For example, like whether or not the matrix has an inverse.

Identity Matrix | Concept

The identity matrix is a function that when given a vector as input outputs an identitical vector as ouput.

Image | Concept

The image of a mapping between two vector spaces is the set of all vectors that map to a non-zero vector.

Kernel | Concept

The kernel represents the set of vectors that map to the zero vector when a matrix is applied to a vector.

Matrix | Concept

In math, a matrix is a function that maps between two vector spaces. A matrix can also be thought of as a shorthand way to write a system of linear equations.

Polynomial Interpolation | Concept

Polynomial interpolation is the process of approximating an unknown function by fitting a polynomial curve to data.

Vector | Concept

A vector has direction and magnitude.

Vector Space | Concept

A vector space is ...

## Examples

2D Rotation Matrix | Example

This example shows how a matrix can be used to rotate a point from one location to another.

2D Scalar Matrix | Example

This example shows how a matrix can be used to scale a point from one location to another.

Cross Product | Example

The cross product of two vectors can be visualized as the area of the parallelogram formed by the two vectors.

Determinant of Three by Three Matrix | Example

The formula for the determinant of a three by three matrix.

Determinant of Two by Two Matrix | Example

The formula for the determinant of a two by two matrix.