Linear Algebra: For f, g C[0,1], let <f,g>=(integral from 0 to 1) f(x)g(x) dx...

[boazuw]

Linear Algebra: For f, g C[0,1], let <f,g>=(integral from 0 to 1) f(x)g(x) dx...

I've been having trouble with a short problem in linear algebra and I'm just not too sure on what it's asking for, or how to use the conditions given.

"For f, g C[0,1], let <f,g>=(integral from 0 to 1) f(x)g(x) dx.
Consider the subspace P3 of polynomials of degree 3 or less with this inner product and describe the "unit sphere" in this subspace i.e. all vectors p within P3 such that ||p||=1."

Any help would be greatly appreciated

(Apologies for the crude formatting of my question: I'm not too sure on how to type out the specific symbols on the forum)

 

[barrick]

I've been having trouble with a short problem in linear algebra and I'm just not too sure on what it's asking for, or how to use the conditions given.

"For f, g C[0,1], let <f,g>=(integral from 0 to 1) f(x)g(x) dx.
Consider the subspace P3 of polynomials of degree 3 or less with this inner product and describe the "unit sphere" in this subspace i.e. all vectors p within P3 such that ||p||=1."

Any help would be greatly appreciated

(Apologies for the crude formatting of my question: I'm not too sure on how to type out the specific symbols on the forum)

Hi boazuw,

You should use the definition: this is the set of polynomials \(\displaystyle f(x)\) such that:

\(\displaystyle \displaystyle\int_0^1 (f(x))^2\;dx=1\)
​

If you write the general polynomial as:

\(\displaystyle f(x) = ax^3 + bx^2 + cx + d\)
​

and compute the integral, you can express this as an equation on the coefficients.

By the way, to format mathematical expressions in this site, you can enclose LaTeX code between the tags [ t e x ] and [ / t e x ] (without the spaces). Try quoting this answer to see an example.