# Linear Transformation

• 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.