16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 3.6.3

Linear Transformations

Mathematics — Linear Algebra · dual-basis-functions-on-set

Problem

Let

S

be a set,

F

a field, and

V(S; F)

the space of all functions from

S

into

F

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

(cf)(x) = cf(x).

Let

W

be any

n

-dimensional subspace of

V(S; F)

. Show that there exist points

x_1, \ldots, x_n

S

and functions

f_1, \ldots, f_n

W

such that

f_i(x_j) = \delta_{ij}