Explain why each of the following subsets is or is not a vec

Explain why each of the following subsets is, or is not, a vector subspace. (No credit will be given without explanation/proof)

(c) The span of the set of functions {1,e^x,e^-x}.

Solution

a)

Span is set of all linear combinations

a+be^x+ce^{-x}

a,b,c are real numbers

The three vectors are linearly independent so we can represent any vector in span as (a,b,c) where a is coefficient w.r.t. 1, b w.r.t. e^x ,c w.r.t. e^{-x}

1. Let, (a,b,c) and (e,f,g) be in the set

So,(a,b,c)+(e,f,g)=(a+e,b+f,c+g) is also in the set

2. Let, (a,b,c) be in the set and x be any scalar

x(a,b,c)=(xa,xb,xc) is also in the set

HEnce the set forms a subspace