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x = (5; 6) and y = (-1; 0), vectors in R^2. Find the unique basis C = {w1;w2} of R^2 with respect to which [x]C = (-1, 2) and [y]C = (2, -1).
I know how to find [x] and [y] when given x, y and w's, but I dont know how to work backwards to find w1 and w2.
When we write a vector, v, in terms of the basis (w1, w2), we are writing it as v= aw1+ bw2 for some numbers a and b. You want x= -1w1+ 2w2= (5, 6) and y= 2w1- w2= (-1, 0).
If you let w1= (a, b) and w2= (c, d) then that says that -1(a, b)+ 2(c, d)= (-a+2c, -b+ 2d)= (5, 6) and 2(a, b)- (c, d)= (2a- c, 2b- d)= -1, 0). That gives you four equations for a, b, c, and d. (Actually, two equations for a and c and two equations for b and d.)