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October 25th, 2017, 04:40 AM   #1
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properties of inner product space

hello everyone!

i am having a question and a problem regarding the properties of an inner product space.

the question is:
if $\displaystyle a= \begin{pmatrix}4&1\\ 1&5\end{pmatrix}$. then if we say v=(y1,y2),u=(x1,x2), the inner product in the standart base based on A would be (u,v)=4x1y1+x1y2+x2y1+5x2y2 ? i get that, but i don't know if it's okay. do you get that as well?

and i don't understand something regarding the properties of an inner product. i'll try to write it as a question:
if we say that ( , ) is an inner product in V over C, then can we say that every product of ( , ) in a scalar $\displaystyle 0 \neq k \in C$ is an inner product in V? by the properties of inner product space, it seems to be true and it does keep the linearity properties. what do you think?
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October 25th, 2017, 05:43 AM   #2
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I find this all very difficult to understand! An inner product on a vector space, V, is a function from VxV to a field, usually the real numbers or the complex numbers. That is, it assigns a number to every pair of vectors subject to the rules
It is linear in the first term: (au+ bv, w)= a(u,w)+ b(v, w)
It is "symmetric": (u, v)= (v, u)* where * is complex conjugation. In particular if the vector space is over the real numbers, (u, v)= (v, u).

(u, u) is ever negative and is 0 only if u= 0.

Now, if we are working with a two dimensional vector space over the real numbers, then we can say that, for any invertible, symmetric, 2 by 2 matrix, $\displaystyle \begin{pmatrix}a & b \\ c & e \end{pmatrix}$, $\displaystyle \begin{pmatrix}x_1 & y_1\end{pmatrix}\begin{pmatrix}a & b \\ c & d \end{pmatrix}\begin{pmatrix}x_2 & y_2\end{pmatrix}$ is an inner product- though that is a very limited idea of "inner product".

Finally, I don't know what you mean by "every product of ( , ) in $\displaystyle k\ne 0 \in C$". Where did "k" come from? What does it mean to say that a product is "in" a complex number?
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October 25th, 2017, 02:06 PM   #3
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Hello Country Boy.

basically, i've asked 2 questions: 1) regarding the properties of inner product space to help me understand it's idea and the 2)was a mathmatical question i've tried to solve.

2)if [math]a=\displaystyle \begin{pmatrix}4 & 1 \\ 1 & 5 \end{pmatrix}[\math] then the inner product that is determined (assigned using A) by a in standard base R^2 is: (u,v)=4x_1y_1+x_1y_2+x_2y_1+5x_2y_2 (v=(y_1,y_2), u=(x_1,x_2). my english is not so good, please try to understand what i'm refering to. i'm doing my best to describe.

1)the theoritical question is if ( , ) (i.e any (u,v)) is an inner product in V over C, so every product of ( , ) (any product of any (u,v)) in a scalar that is not zero, i.e every 0 \neq k \in C (k is a real number), is also an inner product in V. (judging by the basic properties of inner product space, this sentence seems to be correct).

please help me if you can, and sorry for my bad english. doing my best to learn english and math at the same time.
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October 25th, 2017, 04:14 PM   #4
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Your English is far better than my (put pretty much whatever language you like here)!

Yes, that's basically what II said before. For any vectors $\displaystyle \begin{bmatrix}x_1 & y_1\end{bmatrix}$ and $\displaystyle \begin{bmatrix}x_2 & y_2 \end{bmatrix}$ we can define an inner product by $\displaystyle \begin{bmatrix}x_1 & y_1\end{bmatrix}\begin{bmatrix}4 & 1 \\ 1 & 5\end{bmatrix}\begin{bmatrix}x_2 \\ y_2\end{bmatrix}$$\displaystyle = \begin{bmatrix}4x_1+ y_1 & x_1+ 5y_1\end{bmatrix}\begin{bmatrix}x_2 \\ y_1\end{bmatrix}$$\displaystyle = 4x_1x_2+ x_2y_1+ x_1y_2+ 5y_1y_2$.

"1)the theoretical question is if ( , ) (i.e any (u,v)) is an inner product in V over C, so every product of ( , ) (any product of any (u,v)) in a scalar that is not zero, i.e every 0 \neq k \in C (k is a real number), is also an inner product in V."
Sorry but I still don't understand this. First, you are asserting that ( , ) is an inner product but then you appear to be asking if ( , ) is an inner product. Perhaps you are wondering if (u, v) for specific vectors u and v is called an "inner product"? It can be called "the inner product of u and v" but the result is a number, not "an inner product". That is the name of the operation, not of the result. Its like saying that [math]f(x)= x^2[/tex] is a function but the result of that function is a number, not a function.

You say that "any (u, v) is a scalar that is not 0". No, it is quite possible for the inner product of two vectors to be 0.
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November 8th, 2017, 07:30 AM   #5
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right, i understand, thank you. the result of an inner product V over C against a scaller cannot be an inner product, because it is a scaller.

thank you so much!
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