Find the coordinate vector of w 75 relative to the basis u1

Find the coordinate vector of w = (7,5) relative to the basis {u_1 = (2, -4), u_2 = (3,8)} for R^2.

Solution

  Let the desired coordinate vector be ( x,y). Then x,y satisfy v = xu₁ + yu₂ . Then, 2x + 3y = 7 …(1) and -4x + 8y = 5…(2). To solve these equations multiply the 1^st equation by 2 and then add it to the 2^nd equation so that 4x + 6y – 4x + 8y = 14 + 5 or, 14y = 19 so that y = 19/14. On substituting y = 19/14 in the 1^st equation, we get 2x + 3*19/14 = 7 or, 2x = 7 – 57/14 = - 41/14 . Therefore, x = - 41/28. Thus the coordinate vector of (7,5) relative to the given basis { u₁ , u₂} for R² is ( - 41/28, 19/14).