Question

Let S be the standard basis for R^2 and let B={v_1,v_2} be the basis in which v_1=(2,1) and v_2=(-3,4)

Answer 1

how do find the transition matrix P_B-->S

Answer 2

from inspection I don't understand why it's [2 -3; 1 4]

Answer 3

dont you need to go like [1 0 | 2 -3; 0 1 | 1 -1], then try and row reduce until the right side is an identity matrix

Answer 4

Columns of the base matrix are coordinates of of vectors with respect to the standard basis and this matrix is called the transition matrix from a basis to the standard basis.

Going in the other direction u need the inverse matrix( which can be computed using row reduction).