# g is a linear combination of f1, f2,...,fn iff g(N) = {0}

Functional Analysis

Linear Functionals

Vector Space

Suppose that Ɛ is a vector space, nЄN, and f1, f2,...,fn, and g are linear functionals on Ɛ.

Let N = ker(f1)∩ker(f2) ∩...∩ker(fn). Then g is a linear combination of f1, f2,...,fn if and only if g(N) = {0}.

Hint: Suppose that g(N) = {0}. Define ĝ: Ɛ /N →C by ĝ([x]) = g(x) for each x in Ɛ and h: Ɛ /N→Cn by

h([x]) = (f1(x), f2(x),..., fn(x)) for each x in Ɛ. Hence h is one-to-one and onto M = range (h). We have h - 1 M Ɛ /N . Let f = ĝo: h - 1: M→C. The f is a linear functional on M . Extend f to Cn.

See the attached file.

© BrainMass Inc. brainmass.com December 24, 2021, 5:15 pm ad1c9bdddfhttps://brainmass.com/math/combinatorics/linear-combination-38138

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Functional Analysis

Linear Functionals

Vector Space

Suppose that Ɛ is a vector space, nЄN, and f1, f2,...,fn, and g are linear functionals on Ɛ.

Let N = ker(f1)∩ker(f2) ∩...∩ker(fn). Then g is a linear combination of f1, f2,...,fn if and only if g(N) = {0}.

Hint: Suppose that g(N) = {0}. Define ĝ: Ɛ /N →C by ĝ([x]) = g(x) for each x in Ɛ and h: Ɛ /N→Cn by

h([x]) = (f1(x), f2(x),..., fn(x)) for each x in Ɛ. Hence h is one-to-one and onto M = range (h). We have h - 1 M Ɛ /N . Let f = ĝo: h - 1: M→C. The f is a linear functional on M . Extend f to Cn.

See the attached file for the solution of the problem.

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© BrainMass Inc. brainmass.com December 24, 2021, 5:15 pm ad1c9bdddf>https://brainmass.com/math/combinatorics/linear-combination-38138