(A)(a) Let x = u1 + u2 + u3 + u4 Express

Question

(a)

(a) Let x = u1 + u2 + u3 + u4. Express

A18x as a linear combination of u1,u2,u3,u4. You are allowed to have exponents of numbers in your answer. (Hint: x has been expressed as the sum of eigenvectors.)

(b) What are the eigenvalues for A2 −2A?

(c) Let T(x) = Ax, give two vectors v,w such that the triangle with vertices {T(0),T(v),T(w)} has 6 times the area as the triangle with vertices {0,v,w}. Be sure to justify your answer. (Hint: It is unnecessary to compute the area of these triangles.)

(d) What is the general solution to Ax = u2 + 2u3?

Let 37 253 3 ‘21 E W _ —5
A‘ 0 0 2 g
0 0 0 3 be a matrix which decomposes as A = U DU‘I, where 1 —1 18 1 O
0 2 1 —1 0
U_ 0 0 3 7 ’ D_ 0
0 0 0 5 0 Let 114, 112,113, U4 be the columns of U and B = {U1,U2,U3,U4}. CONDO OM00 WOOD
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