Let V be a vector space with basis S TV1 v2 vn Let c e R c 0

Let V be a vector space with basis S TV1, v2, vn). Let c e R, c 0. (a) Show that Si V2, v1, v3, v4, vn is a basis for V (b) Show that S2 tcv1, v2, v3, vn is a basis for V (c) Show that S V1 cv2, v2, v3, vn is a basis for V (Recall that B ui, 2, un t is a basis for V if and only if the following 2 conditions hold 1. B spans V: If v E V, then we can find c1, C ...,cn ER such that 2. B is linearly independent: If c1u1 c2u2 cnun 0, then c1 30, c2 0,

Solution

a) Since S=S1 it follows that S1 also forms a basis.

b) Since v1, v2....vn are linearly independent

cv1, v2, v3....vn also are linearly independent and has dimension n.

Hence S2 also forms a basis.

c) Consider 3 elements in S3

v1+cv2 -cv2 = v1

Or v1+cv2 and v2 are linearly independent.

Hence the whole set is linearly independent and can form a basis