Question
Find the orthogonal projection of $f$ onto $g .$ Use the inner product in $C[a, b]$$$\langle f, g\rangle=\int_{a}^{b} f(x) g(x) d x$$$$C[-\pi, \pi], \quad f(x)=\sin 2 x, \quad g(x)=\cos 2 x$$
Step 1
By definition, this is given by the integral of $f(x)g(x)$ over the interval $[-\pi, \pi]$. So we have $$\langle f, g\rangle=\int_{-\pi}^{\pi} f(x) g(x) dx = \int_{-\pi}^{\pi} \sin(2x) \cos(2x) dx.$$ Show more…
Show all steps
Your feedback will help us improve your experience
Ahmad Reda and 65 other Calculus 3 educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
Find the orthogonal projection of $f$ onto $g .$ Use the inner product in $C[a, b]$ $$\langle f, g\rangle=\int_{a}^{b} f(x) g(x) d x$$ $$C[-\pi, \pi], \quad f(x)=x, \quad g(x)=\cos 2 x$$
Inner Product Spaces
Find the orthogonal projection of $f$ onto $g .$ Use the inner product in $C[a, b]$ $$\langle f, g\rangle=\int_{a}^{b} f(x) g(x) d x$$ $$C[-\pi, \pi], \quad f(x)=x, \quad g(x)=\sin 2 x$$
Find the orthogonal projection of $f$ onto $g .$ Use the inner product in $C[a, b]$ $$\langle f, g\rangle=\int_{a}^{b} f(x) g(x) d x$$ $$C[-\pi, \pi], \quad f(x)=\sin x, \quad g(x)=\cos x$$
Transcript
18,000,000+
Students on Numerade
Trusted by students at 8,000+ universities
Watch the video solution with this free unlock.
EMAIL
PASSWORD