| 13 Define T: R3 → R2 by T(x, y, z) = (4x + 5y + 6z, 7x +8y + 9z) Suppose o1,02 denotes the dual basis of the standard basis of R2 and V1, 2, V3 denotes the dual basis of the standard basis of R3. Describe the linear functionals T' (@1) and T' (02). (a) Write T'(1) and T'(42) as linear combinations of 1, 2, V3. (b)

The matrices A₁ = A3 [] = ^--^- A₂ = [] =[] A4 form a basis for the linear space V = R 2x2. Write the matrix of the linear transformation T: R 2x2R 2x2 such that T(A) = 3A + 10AT relative to this basis.

4 Let W = span(vV1, V2) where vi = and v2 2 . Then: (a) Find the orthogonal decomposition of the vector x = 3 (b) Find an orthogonal basis for W- (remember to justify your answer).

The vectors p1(t) = 1 + t², p2(t) = −t + 3t², and p3(t) = 1+t+t2 form a basis B 2(t), p3(t)} for P₂(R). Let p(t) = 1 + 2t + 4t², find its B-coordinates [p(t)B. =

Identify all the polynomial functions of two variables of degree < 2 which are harmonic. Show that they form a vector space of dimension 5 and give a vector basis of that space.