Consider the basis v1 v2 for R2 where v1 1 1T and v2 23T S

Consider the basis [v_1, v_2] for R^2, where v_1 = (1, 1)^T and v_2 = (2,3)^T. Suppose v = c_1v_1 + c_2v_2 for scalars c_1, c_2. What are the coordinates of v with respect to the standard basis? Now suppose v = (d_1, d_2)^T = d_1 e_1 + d_2 e^2 where e_1 and e_2 are the standard basis vectors. What are the coordinates of v with respect to v_1 and v_2?

Solution

a) v = c1 (v1) + c2 (v2) = c1 (1 ,1 ) + c2 (2,3) = (c1 +2c2 ,c1 +3c2)

standard basis (1,0) ,(0,1)

so v =    (c1 +2c2) (1,0) + (c1 +3c2) (0.1) ,so coordinates of v with respect to standard basis are

(c1 +2c2 ,c1 +3c2)

b) v = (d1,d2) = a(v1) + b (v2)

(d1,d2) = a(1,1) + b(2,3)

d1 = a + 2b

d2 = a + 3b

   a = (3d1 - 2d2 ) b = (d2 - d1)

so coordinate of v with respect to v1 and v2 = ((3d1 - 2d2 ), (d2 - d1) )