Share
Explore BrainMass

Continuous, Real Valued, Linearly Dependent Functions and Matrix Determinants

Let f1,...,fk be continuous real valued functions on the interval [a,b]. Show that the set {f1,...,fk}is linearly dependent on [a,b] iff the k x k matrix with entries
b
<fi,fj> = &#8747; fi(x)fj(x)dx has determinant zero.
a

See attached file for full problem description.

Attachments

Solution Preview

Part 1: proof of "only if"

Here we prove that if {f_1, ...f_k} are linearly dependent, det( <f_i, f_j> ) = 0.

The linear dependence means that one of the k functions, say f_m, is a linear combination of the rest of the functions, that is f_m(x) = sum_j a_j f_j(x), where a_j are some coefficients.
If so, the row
<f_m, f_i>, i = 1,...,k
is a linear combination of the rest of the rows in the matrix <f_i, f_j>:
<f_m, f_i> = sum_j a_j <f_j, f_i>
and therefore, by the properties of determinants, det( <f_i, f_j> ) = 0.

Part 2: proof of "if"

Here we ...

Solution Summary

Continuous, Real Valued, Linearly Dependent Functions and Matrix Determinants are investigated.

$2.19