- Transformations
- Onto
- In Transformations
- Theorem for 1-1
- Theorem for Onto
- Example
- Linear Transformations
- Example
- Find the Matrix
- Example
- Rotation Transformation by any ⁍

# Transformations

## Onto

is onto if Range =

For each in , the equation __has at ____least ____one solution__

## In Transformations

A transformation T: → is...

‣

**One-to-one**

or injective, if for every in , the equation __has at ____most ____one solution__

- no two different inputs have the same output
- if then
- horizontal line test
- we can solve it, our solution is the only one
- ex where has a pivot in every column

‣

**Onto**

if Range

- we can always solve it
- ex where has a pivot in every row

## Theorem for 1-1

A matrix , →

The following are equivalent

- is 1-1
- for each in , has at most one solution
- for each in , has at either no or exactly 1 one solution
- has only the trivial solution
- The columns of are linearly independent
- A has a pivot in every column

## Theorem for Onto

A matrix , →

- is onto
- is consistent for each in
- is consistent for each in
- has a pivot in each row

if is , then pivot in each row ↔ pivots ↔ pivot in each column
onto ↔ is 1-1

## Example

Not 1-1, but is onto (pivot in each row)

Nul = Span ,

# Linear Transformations

Transformation islinearif for all in and scalars :

these properties are the exact same as the linearity properties for matrices because transformations and matrix transformation are the same thing

If is linear, then

## Example

Linear or not?

No,

## Find the Matrix

Given a linear transformation , how do we find so that ? (Call A the standard matrix for )

### Example

Find the standard matrix for