I'm learning Euclidean geometry, specifically inner products and Euclidean spaces, and need help with the following exercise:
In the vector space $\mathbb R^2$, we write $\mathbf x = (x_1, x_2), \mathbf y = (y_1, y_2)$ ect. for points $\mathbf x, \mathbf y$ in terms of a given base. For simplicity we assume that we are working over the field $\mathbb R$.
Which of the following bilinear, symmetric functions $f(\mathbf x, \mathbf y)$ define an inner product?
$(1) \quad f = 2x_1y_1 - 3x_1y_2 - 3x_2y_1 + 3x_2y_2$
$(2) \quad f = 3x_1y_1 - 2x_1y_2 - 2x_2y_1 + 4x_2y_2$
Give the norm which is defined by the inner product (when it's defined). Describe the curves $f(\mathbf x, \mathbf x) =$ constant in each case.
First of all, I don't understand what it means for $f$ to be symmetric. Does this mean that I can interchange the $x_i$ and $y_i$ in each of the linear term of $f$?
I did go back several times to the definition of an inner product but it is still unclear to me how I should solve this problem. I wanted to show that the $3$ axioms defined by an inner product are satisfied (or not), i.e. the symmetry $<\mathbf x, \mathbf y> = <\mathbf y, \mathbf x>$ (since we are working over the field $\mathbb R$ and not $\mathbb C$), the linearity in the first component and the positive definiteness.
For the symmetry, maybe because the function $f(\mathbf x, \mathbf y)$ is symmetric I can write
$$f(\mathbf x, \mathbf y) = <\mathbf x, \mathbf y> = 2x_1y_1 - 3x_1y_2 - 3x_2y_1 + 3x_2y_2 = 2y_1x_1 - 3y_2x_1 - 3y_1x_2 + 3y_2x_2 = <\mathbf y, \mathbf x>$$
but I'm not sure if this is any good. I would also appreciate any help on the $2$ sub-questions.
The bilinear form $\;f\;$ is symmetric if $\;f(\vec x,\vec y)=f(\vec y,\vec x)\;$ .
In the given example, you have to check symmetry as defined above. For example with the first one:
$$f((x_1,x_2),\,(y_1,y_2))=2x_1y_1-3x_1y_2-3x_2y_1+3x_2y_2\stackrel?=$$
$$\stackrel?=2y_1x_1-3y_1x_2-3y_2x_1+3y_2x_2=f((y_1,y_2),\,(x_1,x_2))$$
Check carefully the above: does that equality there with the ? sign is true? Then the form is symmetric. Otherwise, it is not.
Now you try this again also with the second given form