Linear Transformation

A Linear transformation is a Transformation where

• where
• Domain of T is (where we start)
• Codomain or target of T is
• The vector is the image of under T
• The set of all possible images is called the range
• image range codomain
• When the domain and codomain are both , you can represent them as a Cartesian Graph in , as in a mapping of
  • If y is the codomain and x is the domain, the range is the range, the domain is all the images of f(x)
• Addition Rule
  • T(u + v) = T(u) + T(v)
• Multiplication Rule
  • T(c) = cT(v)
• Prove a transformation is linear by proving the addition and multiplication rules.
• “Principle of Superposition”
  • If we know , …, then we know every T(v)

1-1

Injective

If there is at most one location in the Codomain for every location in the Domain

Link to original

• 1-1 every column of T is a pivot column
• 1-1 iff standard matrix has pivot in every column

Onto

Surjective

If there is a location in the Codomain for every location in the Domain

Link to original

• TLDR: Onto every row of T is a pivot row
• onto iff the standard matrix has a pivot in every row
• The matrix A has columns which span .
• The matrix A has all pivotal rows.

Bijective

Bijective

If there is exactly one location in the Codomain for every location in the Domain for the Transformation

Link to original

• Needs square matrix

Matrix Multiplication is a Linear Transformation

Compute

Calculate so that

Give a Give a that is not in the range of Give a that is not in the span of the columns of (Same question for all)

Range of is a bunch of images of the following form:

For to not be in the range of , it cannot be in the above form, e.g. it can be