Matrix Multiplication is a Linear Transformation

Timmy is Great

When I first met matrix, I was totally confused. After learning LAFF and 3blue1brown course, and I finally have more confidence with Matrix.

Linear transformation in LAFF

$$ L(\alpha x) = \alpha L(x) $$ $$L(x+y)=L(x) + L(y)$$

or(iff) $$L(\alpha u+ \beta v) = \alpha L(u) + \beta L(v)$$

Linear transformation in 3blue1brown Course

after linear transformation(intuitive animations):

  1. Origin remain fixed.
  2. Lines remain lines.
  3. Grid lines remain parallel and evenly spaced.

Every column of a matrix is a transformation of its basis vector, combine the effects of all the columns(transformations of all basis vectors), that is the transformation effect of the whole matrix.

Timmy is a great tool, that allow us to manipulate linear transformation intuitively and instantly.