Move the blue points to see the effects of a linear transformation in R^2.
e_1' and e_2' = where the standard basis vectors e_1 and e_2 are transformed.
The matrix of the transformation will have columns e_1' and e_2'.
OLD = the point you want to transform.
The new point is NEW = T(OLD).
The red arrows show that we move along e_1' and e_2' distances corresponding to the x- and y-coordinates of OLD.
A, B, C, D = four points that determine a quadrilateral; move them around to see how the transformation changes its shape (transformed points A', B', C', D' form a new quadrilateral).