find t to have an inner product

[Ola Halawi]

For which values of \(\displaystyle t\) is

\(\displaystyle \langle p,\, q \rangle\, =\, \displaystyle{\int}\, 2t p(x) q(x)\, dx\)

an inner product on \(\displaystyle V\,=\,P_2\)?

I tried to use the properties of an inner product such that \(\displaystyle \langle p(x),\,p(x) \rangle \,=\,0\) if and only if \(\displaystyle p(x)\,=\,0\), but the equation turned out to a mess and I could not find \(\displaystyle t\). Any help would be much appreciated.

 

[pka]

For which values of [FONT=MathJax_Math]t[/FONT] is

[FONT=MathJax_Main]⟨[/FONT][FONT=MathJax_Math]p[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math]q[/FONT][FONT=MathJax_Main]⟩[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Size2]∫[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Math]t[/FONT][FONT=MathJax_Math]p[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Math]q[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Math]d[/FONT][FONT=MathJax_Math]x [/FONT]an inner product on [FONT=MathJax_Math]V[FONT=MathJax_Main]=[FONT=MathJax_Math]P[/FONT][FONT=MathJax_Main]2[/FONT][/FONT][/FONT]? I tried to use the properties of an inner product such that [FONT=MathJax_Main]⟨[FONT=MathJax_Math]p[FONT=MathJax_Main]([/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math]p[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]⟩[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]0[/FONT][/FONT][/FONT] if and only if [FONT=MathJax_Math]p[FONT=MathJax_Main]([FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]0[/FONT][/FONT][/FONT], but the equation turned out to a mess and I could not find [FONT=MathJax_Math]t[/FONT]. Any help would be much appreciated.​

Please check your posting.
A inner product is a function from \(\displaystyle \mathcal{V}\times\mathcal{V}\to\mathbb{R}\).
So you need the integrals to have limits of integration.
​

 

[Ola Halawi]

Sorry there is a typo the integral is from t to 2 and i should find t

 

[Ola Halawi]

pleaaaaaaaaaaaaaaaaaaaaaase i need an answer to this question, it is tooooooooo urgent.

 

[pka]

Sorry there is a typo the integral is from t to 2 and i should find t

Lets be clear the inner product is \(\displaystyle \displaystyle \left\langle {p,q} \right\rangle = \int_t^2 {p \cdot q} \).

Where \(\displaystyle \mathcal{V}=P_2=\{ax^2+bx+c: \{a,b,c\}\subset\mathbb{R}\}\)

 

[pka]

pleaaaaaaaaaaaaaaaaaaaaaase i need an answer to this question, it is tooooooooo urgent.

Please never NEVER PM to beg for help! You did not deem it proper to answer my questions.
If I guessed correctly then let \(\displaystyle t=0,~p(x)=x-1~\&~q(x)=1\).
Now clearly \(\displaystyle <p,q>=0\). So that violates the zero property.

The zero property is, I think, the difficulty with this definition of the inner product.