For Problems a to b, suppose that T: V → W is a linear transformation from vector space V to vector space W. Also suppose that w is a solution of T(u) = b1, and that u1 is a solution of T(u) = b2. Then u1 + u2 is a solution of T(u) = b1 + b2; this is called the Superposition Principle. as first introduced in Sec. 2.1.
a. Use linearity to prove the Superposition Principle.
b. Show that y = cost - sin t is a solution of the non homogeneous linear equation y" - y' - 2y = 4 sin t - 2cos t.