Problem

Line of Solutions If and are vectors in vector space , with , then the line through in...

Line of Solutions If and are vectors in vector space , with , then the line through in the direction is defined to be the set .

(a) Find the line in ℝ2 through in the direction .
(b) Find the line in ℝ3 through in the direction .
(c) Show that solutions of y′ + y = 0 are a subspace of , and that every vector in this subspace is a multiple of e−t.
(d) Show that the solutions of y′ + y = t form a line in through t − 1 in the direction e−t.
(e) Relate parts (c) and (d) to what you learned about solutions to homogeneous and nonhomogeneous differential equations in Sec. 2.1.