5. Vectors and Matricies

What is the general concept of a vector

they can be added together and multiplied by a scalar

4 examples of vectors

1. Geometric vector

2. Polynomial

3. Audio signals

4. Elements of

what is the difference between elements of Rⁿ and geometric vectors

elements of Rⁿ do not need geometry to exist, geometric vectors can only exist in R² or R³

transpose of a matrix

swap rows and columns

trace of a matrix

sum of the main diagonal

relationship between the trace of a matrix and its eigenvalues

the trace is equal to the sum of eigenvalues

Why are audio signals vectors

Audio signals are represented as a series of numbers, they can be added and scaled to give another audio signal

Why are polynomials vectors

They can be summed and scaled to give another polynomial

Definition of a matrix

With M, N ∈ N a real-valued (M,N) (or M ×N) matrix A is an M·N-tuple of elements Aij, i∈{1,...,M}, j ∈{1,...,N}, which is ordered according to a rectangular scheme consisting of M rows and N columns:

For (M x N) what represents the column and row

M is the row

N is the column

Sum of two matrices is defined as

The elementwise sum

Product of matrices

Identity matrix

Leading diagonal 1 rest of the elements 0

General system of linear equations with N equations and N unknowns

A₁,₁x₁ +...+ Aₙ,₁xₙ = b₁

…

Aₙ,₁x₁ +...+ Aₙ,ₙxₙ = bₙ

Ax = b