16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 8.1.12

Inner Product Spaces

Mathematics — Linear Algebra · inner-products

Problem

Let

V

be a finite-dimensional vector space and let

\mathfrak{B} = \{\alpha_1, \ldots, \alpha_n\}

be a basis for

V

. Let

( \ | \ )

be an inner product on

V

. If

c_1, \ldots, c_n

are any

n

scalars, show that there is exactly one vector

\alpha

V

such that

(\alpha|\alpha_j) = c_j

j = 1, \ldots, n