Question

3. (10 points) Let T : R3-A, T(x1, 22, 23) = (zi,-z2,23 a) Prove that T is a linear operator. b) Find the standard matrix of T.

Answer 1

T: R^3 rightarrow R^3 such that: T(x_1, x_2, X_3) = (x_1, x_2, x_3) a) To prove: rightarrow T is Linear operation. i.e., we have to prove: (I) T (x + y) = T (x) + T(y) (II) (T alpha x) = alpha T(x) where, x, y elementof R^3 i.e., x = (x_1, x_2, x_3) y = (y_1, y_2, y_3) elementof alpha elementof R Now: (1) T((x_1, x_2, x_3) + (y_1, y_2, y_3)) T = (x_1 + y_1, x_2 + y_2, x_3, + y_3) = (x_1 + y_1, - (x_2 + y_2), x_3 + y_3) (by definition of T) = (x_1 + y_1, - x_2 - y_2 - x_3, + y_3) (x_1 - x_2, x_3) + (y_1, - y_2, y_3) \r\n = T(x_1, x_2, x_3) + T(y_1, y_2, y_3) Rightarrow T (x + y) = T (x) + T(y) (II) T (alpha x) = T(alpha (x_1, x_2, x_3)) = T(alpha x_1, alpha x_2, alpha x_3) = (alpha x_1, - (alpha x_2), alpha x_3) = alpha (x_1, - x_2, x_3) = alpha T(x_1, x_2, x_3) = alpha T (x) Therefore (I) and (II), T is linear operation b) standard Basis of R^3: B{(1, 0, 0), (0, 1, 0), (0, 0, 1)}