4. (a) Show that the mapping T: R² R² given by T(x, y) = (y, ry) is not a linear

mapping.

(b) Let a: C³

C³ be the mapping given by

a(x, y, z) = (x+2y+z, 2x − y + z, 4x + 3y +3z).

i. Prove that a is linear.

ii. Find a basis for the kernel of a.

iii. Find the rank and nullity of a.

iv. Verify that the Rank-Nullity Theorem holds for this map.

(c) Suppose A is a real symmetric matrix with eigendecomposition QDQT, where Q

is orthogonal and D is diagonal.

i. Prove that A" = QD"QT for all n N = {1,2,3...}.

ii. Prove that if the eigenvalues of A are all non-zero, then A-¹ =

QD-¹Q™.