LinearAlgebra.md

Linear Algebra

Linear Equations

Structure

One of the most basic motivations towards making computers was solving a system linear equations given by

where , , ... are unknowns and 's and 's denote the constants, either complex or real.

Definitions

A system is called a solution of the above system of linear equations if satisfies each of the eqations in our system of linear equations simultaneously and when , then the system of equations is called to be homogeneous. Such system of equations have one certain solution = , = , ... = called the trivial solution. A system of Linear equations is said to be consistent if it has at least one solution and inconsistent if it has no solutions.

Vectors

A vector is a numeric measurement with directions. In two dimensions, a vector is of the form and the magnitude of this vector is given by

Vector Multiplications

Dot Product

There are two ways to go about this:

1. 

   This is the summation of element wise multiplication of the two vectors. The notation denotes that the vectors are column vectors and the result of the equation above would be a 1x1 vector which is a scalar quantity.

2. 

This notation is not very convenient for vector multiplication unless a the angle on the right hand side is known to us. Although, it is a much more common practice to use this equation for finding out the angle between two vectors using

Outer Product

The outer product of two vectors results in a matrix and is given by the equation:

If there are two column vectors u1 and v1 that are given by and respectively. Then their outer product is written as

which then results to be .

Matrices

Matrices are two dimensional set of numbers. These are very efficient data types for quick computations.

Matrix Multiplications

Dot Product

The product of two matrices A and B in given by the formulae