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

Not what you're looking for?

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.

##### Purchase this Solution

##### Solution Summary

This solution is comprised of a detailed explanation of the normed linear spaces.

It contains step-by-step explanation for the following problem:

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}.

Solution contains detailed step-by-step explanation.

##### Solution Preview

Functional Analysis

Linear Functionals

Vector Space

...

###### Education

- BSc, Manipur University
- MSc, Kanpur University

###### Recent Feedback

- "Thanks this really helped."
- "Sorry for the delay, I was unable to be online during the holiday. The post is very helpful."
- "Very nice thank you"
- "Thank you a million!!! Would happen to understand any of the other tensor problems i have posted???"
- "You are awesome. Thank you"

##### Purchase this Solution

##### Free BrainMass Quizzes

##### Exponential Expressions

In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.

##### Graphs and Functions

This quiz helps you easily identify a function and test your understanding of ranges, domains , function inverses and transformations.

##### Solving quadratic inequalities

This quiz test you on how well you are familiar with solving quadratic inequalities.

##### Know Your Linear Equations

Each question is a choice-summary multiple choice question that will present you with a linear equation and then make 4 statements about that equation. You must determine which of the 4 statements are true (if any) in regards to the equation.

##### Geometry - Real Life Application Problems

Understanding of how geometry applies to in real-world contexts