- 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.