**Orthogonal Projection of Functions**To find the orthogonal projection of \( f \) onto \( g \), we use the inner product in \( C[a, b] \). The inner product in \( C[a, b] \) is given by the integral:\[\langle f, g \rangle = \int_a^b f(x) g(x) \, dx\]Given:- Interval: \( C[0, 1] \)- Functions: \( f(x) = 2x \) and \( g(x) = e^x \)We calculate the orthogonal projection of \( f \) onto \( g \).The formula for the orthogonal projection \( \text{proj}_g f \) is:\[\text{proj}_g f = \frac{\langle f, g \rangle}{\langle g, g \rangle} g\]Therefore, we need to compute the inner products \( \langle f, g \rangle \) and \( \langle g, g \rangle \).\[\langle f, g \rangle = \int_0^1 (2x) (e^x) \, dx\]\[\langle g, g \rangle = \int_0^1 (e^x) (e^x) \, dx\]Calculating these integrals and substituting back into the orthogonal projection formula will give the result for \( \text{proj}_g f \).

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Answered: Find the orthogonal projection of f…

Find the orthogonal projection of f onto g. Use the inner product in C[a, b] (f₁ g) = f(x f(x)g(x) dx. Ja C[0, 1], f(x) = 2x, g(x) = ex projgf =

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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**Orthogonal Projection of Functions**

To find the orthogonal projection of \( f \) onto \( g \), we use the inner product in \( C[a, b] \). The inner product in \( C[a, b] \) is given by the integral:

\[
\langle f, g \rangle = \int_a^b f(x) g(x) \, dx
\]

Given:
- Interval: \( C[0, 1] \)
- Functions: \( f(x) = 2x \) and \( g(x) = e^x \)

We calculate the orthogonal projection of \( f \) onto \( g \).

The formula for the orthogonal projection \( \text{proj}_g f \) is:

\[
\text{proj}_g f = \frac{\langle f, g \rangle}{\langle g, g \rangle} g
\]

Therefore, we need to compute the inner products \( \langle f, g \rangle \) and \( \langle g, g \rangle \).

\[
\langle f, g \rangle = \int_0^1 (2x) (e^x) \, dx
\]

\[
\langle g, g \rangle = \int_0^1 (e^x) (e^x) \, dx
\]

Calculating these integrals and substituting back into the orthogonal projection formula will give the result for \( \text{proj}_g f \).

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