Linear Algebra

112 Chapter 2. Vector Spaces
� 1.39 (a) Prove that a set of two perpendicular nonzero vectors from R n is linearly
independent when n>1.
(b) What if n =1? n =0?
(c) Generalize to more than two vectors.
1.40 Consider the set of functions from the open interval (−1..1) to R.
(a) Show that this set is a vector space under the usual operations.
(b) Recall the formula for the sum of an infinite geometric series: 1+x+x 2 +···=
1/(1−x) for all x ∈ (−1..1). Why does this not express a dependence inside of the
set {g(x) =1/(1 − x),f0(x) =1,f1(x) =x, f2(x) =x 2 ,...} (in the vector space
that we are considering)? (Hint. Review the definition of linear combination.)
(c) Show that the set in the prior item is linearly independent.
This shows that some vector spaces exist with linearly independent subsets that
are infinite.
1.41 Show that, where S is a subspace of V , if a subset T of S is linearly independent
in S then T is also linearly independent in V . Is that ‘only if’?