matrix transformations and multiplication

transformation from Rⁿ to R^m is a

rule that a given input in Rⁿ has an R^m output

transformation notation

T:Rⁿ → R^m ; Rⁿ is the domain, R^m is the codomain

we define transformation (as notated above) as T_A =

Av, for all v ∈ Rⁿ (domain’s space)

reflection in x-axis

multiply vector/point by

A = [1 0]

[0 -1]

rescaling in x direction (and key c values)

multiply vector/point by

A = [c 0]

[0 1]

x-expansion if c > 1

x-contraction if 0 < c < 1

rescaling in y direction

multiply vector/point by

A = [1 0]

[0 c]

x-shear

multiply vector/point by

A = [1 c]

[0 1]

identity transformation

I_nv = v ; Rⁿ → Rⁿ

zero transformation

0_mxnv = 0→ ; Rⁿ → Rⁿ

can discern matrix transformation if it

sends zero vector to itself

define composition T º S: Rⁿ → R^l by

[T • S][v] = T(S(v)); S is followed by T

is it a given that AB = BA?

no, but is possible

IA =

A (and commutative)

A(BC) =

(AB)C (associative)

A(B + C) =

AB + AC (distributivity)

(B + C)A =

BA + CA

k(AB) =

(kA)B = A(kB)

if AB = CB, we cannot conclude that

A = C

block multiplication

see miguel lecture 7