Proving a function is an inner product

In summary: Therefore,x^2 + 2Bx+C = 0 Now, substituting in the values for x1 and x2,x^2 + 2Bx+C = 0 Therefore, the equation becomes:x^2+2Bx+C = 0If you solve for x, you will find that it is equal to 0.
  • #1
Vespero
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0

Homework Statement



I'm supposed to show that a function is an inner product if and only if b^2 - ac < 0 and a > 0.

I have proven all of the properties except that <x,x> > 0 if x!= 0. I would write the function out, but can't seem to get matrices to work.

The function is the product of a 1x2 matrix with entries x1 and x2, a 2x2 matrix with entries a, b, b, c, (filling in the top row, then moving to the bottom row), and a 2x1 matrix with entries y1, y2.

After multiplication, I arrive at
<x,x> = a(x1)^2 + 2b(x1)(x2) + c(x2)^2 > 0.

Now I must prove that this is true if and only if b^2 - ac < 0 and a > 0, but am not sure how to go about this.


Homework Equations





The Attempt at a Solution



I thought about different ways of inserting the conditions into the inequality or assuming that they were false and trying to arrive at a contradiction, but I can't seem to utilize them in a useful way.
 
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  • #2
Some other things to try are:

Plugging in specific values for x1 and x2 to get a system of inequalities involving only a, b, and c.

Using the power of linear algebra to analyze the matrix and see if you can simply the problem or otherwise transform it into an easier one.

Using the power of high-school algebra to study and simplify a quadratic equation.
 
  • #3
I'm guessing your function is:
[tex]\langle \mathbf{x},\mathbf{y}\rangle = \left( \begin{array}{c c} x_1 & x_2 \end{array}\right) \left( \begin{array}{c c} a & b\\ b & c\end{array}\right)\left(\begin{array}{c} y_1\\y_2 \end{array}\right)[/tex]

I think your best bet is to complete the square on the quadratic. Treat it as a quadratic polynomial in x1.

Recall the process of completing the square:
[tex] x^2 + 2Bx+C = x^2 + 2Bx + B^2 - B^2 + C = (x+B)^2 -B^2 + C[/tex]
 

1. What is an inner product?

An inner product is a mathematical operation that takes two vectors and produces a scalar value. It is similar to a dot product, but it also satisfies certain properties such as symmetry, linearity, and positive definiteness.

2. How do you prove that a function is an inner product?

To prove that a function is an inner product, you must show that it satisfies the four properties of an inner product: symmetry, linearity, positive definiteness, and conjugate symmetry. This can be done by manipulating the function algebraically or using geometric interpretations.

3. What is the importance of proving a function is an inner product?

The concept of an inner product is fundamental in mathematics, especially in the fields of linear algebra, functional analysis, and quantum mechanics. Proving that a function is an inner product allows us to use the powerful tools and theorems that are based on this concept.

4. Can two different functions have the same inner product?

Yes, it is possible for two different functions to have the same inner product. This is because an inner product only depends on the properties of the function and not on the specific values of the vectors. For example, the inner product of two functions may be the same even if one function is shifted or scaled compared to the other.

5. What are some common examples of inner product functions?

Some common examples of inner product functions include the dot product in Euclidean space, the cross product in three-dimensional space, and the standard complex inner product in complex vector spaces. Other examples include the inner product of polynomials, matrices, and functions in function spaces.

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