linear transformations and their matrices

a function is a linear transformation if it

preserves vector addition and scalar multiplication i.e. T(x + y) = T(x) + T(y) and T(cx) = cT(x)

linear operator

T: Rn -> Rn

matrix that induces a transformation

A is [ T(e₁) … T(e_n) ]

matrix of a counterclockwise rotation by angle theta/ clockwise rotation by angle -theta

[cos theta -sin theta]

[sin theta cos theta}

matrix for horizontal shear

[1 -1]

[0 1]

what happens in a horizontal shear

ever point is displaced horizontally by an amount proportional to its y coordinate

projection matrix onto the line spanned by u

P = uu^T/u^Tu

reflection matrix over the line spanned by u

2P - I where P is the projection matrix

a vector w in R^m is said to be hit by T: Rⁿ → R^m if w =

Tv for some v in Rⁿ

the kernel of T is defined by

ker T = { v in Rⁿ such that T(v) = 0 }; the subspace of Rⁿ containing those the map onto the zero vector in R^m

the image of T is defined by

im T = { T(v) such that v is in Rⁿ }; the subspace of R^m that T maps onto

if the m x n matrix A induces the linear transformation T_a(x) = Ax, then imTa =

col A

T is said to be injective/one to one if

T(v) = T(w) implies v = w, so no element of R^m gets hit twice

T is said to be surjective/onto if

im T = R^m, so that every vector in R^m is hit at least once

T is injective iff

ker T = {0}

if T: Rⁿ → R^m, then n =

dim(kerT) + dim(imT)

if T: Rⁿ → Rⁿ

T injective → T surjective and vice versa

properties of injective

solutions to T(x) = b are unique if exist, all columns of its matrix are linearly independent, and ker T = {0}

properties of surjective

solutions to T(x) = b exist for every b in R^m, all rows of its matrix are linearly independent, and image is all of Rm

properties of invertible

combines those of injective and surjective

number of LI columns of A equals

number of LI rows