Linear Algebra Theory (Week 1 to Week 8)

Algebraic Properties of Vectors in Real Numbers

Commutativity, Associativity, additive identity, additive inverse, distributive, associative property of multiplication, multiplicative identity

Commutativity

u + v = v + u

Multiplicative identity

1u = u

Associative property of multiplication

c (du) = (cd) u

additive inverse

u + (-u) = 0

Linear combination of a vector v

A vector v is a leaner combination of vectors v1, v2, ..., vk if there are scalars C1, C2, ..., Ck such that v = C1v1 + C2v2 + ... + Ckvk.

The scalars C1, C2, ..., Ck are called coefficients of the linear combination.

Span the coordinate plan

The set of all linear combicaitons of a list of vectors v1, ..., vm in V is called the span of v1, ..., vm denoted by span( v1, ..., vm).

In other words, span (v1, ..., vm) = {a1v1 + ... + amvm: a1, ..., am are constants}

The span of the empty list () is defined to be {0}.

First four properties of addition for vector spaces

Commutativity, Associative Zero vector, and additive inverse.

Zero vector

if there exists a special vector, denoted by bold 0, such that v + 0 = v for all vectors in V.

Two properties of multiplication for vector spaces

Multiplicative identity, multiplicative associativity

Two properties of multiplication connected to addition for vector spaces

1.) α (v + w) = αv + αw for all v, w in V and all scales α.
2.) ( α + β) v = α v + βv for all vectors in V and all scalars α, β.