# Linear Algebra Index

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

## Notation

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

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

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

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.

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

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

## Formulas

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

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.

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

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

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

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

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

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.

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

## Operators

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

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

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

## Concepts

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.

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

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

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

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

A vector has direction and magnitude.

A vector space is ...

## Examples

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

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

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

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

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