- Visually:

- All lines must remain lines

- The origin must remain fixed in place

- (parallel lines stay parallel and evenly spaced)

- Only need to record how the basis vectors move

- Then new definitions can be described in terms of the new basis vectors

- x * new i-hat + y * new j-hat

- a 2D linear transformation can be described by 4 numbers (new endpoints for i-hat, j-hat)

- Expressed as a matrix, columns are where i-hat and j-hat land

- This becomes a matrix multiplication

- [ [ a b ] [ c d ] ][ [x] [y] ] = [ [ax + by] [cx + dy] ]

- Matrix to the left of the vector, like a function

- If the new vectors are linearly dependent, then the transformation will squish the space.

- Can also be applied to functions

- Additivity: L(v + w) = L(v) + L(w)

- Scaling: L(c v) = c L(v)

- Preserve vector addition and scalar multiplication

- Derivative is linear.

- Basis function can be the powers of x (infinite)

- Can represent the derivative as a matrix multiplication, by taking the derivatives of each basis function per column.