Linear Algebra Change Of Basis: How is the textbook getting these coefficients? We are given basis B = (u1, u2) and basis C = (v1,v2). u1 = (-1,2), u2=(2,-1), v1=(1,0), v2=(1,1) The coordinate vector x with respect to b is [x]B = (1,3) and coordinate vector x with respect to c is [x]C = (6, -1) The example in the textbook asks: Using the bases B and C, find [x]C given that [x]B = (1,3) This is where I get lost: u1 = (-1, 2) = -3(1,0) + 2(1,1) = -3v1 +2v2 u2 = (2,-1) = 3(1,0) - (1,1) = 3v1 - v2 Where are they getting the coefficients -3 and 2 in u1? Where are they getting the coefficients 3 and -1 in u2? They don't explain it anywhere.

In this case they probably just got them out of their head since it's not that difficult to see.

But if you want a methodic approach, just set $$ (-1,2)=\alpha(1,0)+\beta(1,1)\tag{1} $$ with $\alpha,\beta$ to determine. Now $(1)$ is equivalent to $$ -1=\alpha+\beta\\\ 2=\beta $$ Well, you should be able to solve this linear system!

Same thing for $u_2$ obviously.