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I am just doing my homework and I believe that there is a fault in the problem set.

Consider the set of functions defined by

V= f : R → R such that f(x) = a + bx for some a, b ∈ R

It is given that V is a vector space under the standard operations of pointwise

addition and scalar multiplication; that is, under the operations

(f + g)(x) := f(x) + g(x)

(λf)(x) := λf(x).

Consider the set S = {f1, f2} ⊆ V consisting of the vectors

f1 : R → R defined by f1(x) = 1

and

f2 : R → R defined by f2(x) = x.

(a) Show that the set S is a basis of V ; that is, show that S is a linearly independent

set which spans V . Also state the dimension of V .

(b) State the coordinates (f1)S and (f2)S of the vectors f1 and f2 with respect to

the basis S

, what is wrong is that if a set of linear functions is independat that would mean that Af1+Bf2=0 only has teh trivial solution A=B=0 as a solution but this is obviously not right as f1(x)=1 and f2(x)=x so A+Bx=0, has other solutions to it than the trivial, thus its not indpendant and S not a basis of V.