When I read about Bilinear, Quadratic form and Inner product I saw
Bilinear: If $\beta=(b_1,b_2,\cdots,b_n)$ is an ordered basis for a metric vector space $V$, then a bilinear form is completely determined by the $n\times n$ matrix of values $M_{\beta}=(a_{i,j})=\left(⟨b_i,b_j⟩\right)$ This is referred to as the matrix of the form $($or the matrix of V$)$ with respect to the ordered basis $\beta$.
Inner Product: If $⟨x,y⟩$ defines an inner product, then we can write $⟨x,y⟩=x^TAy$ where $A=(a_{ij})$ is symmetric positive definite and $a_{ij}=⟨e_i,e_j⟩$.
Quadratic form: A quadratic form over a field $K$ is a map $q:V\rightarrow K$ from a finite dimensional $K$ vector space to $K$ such that $q(av)=a^2q(v)\forall v\in V,\forall k\in K$ we can write $q(x)=x^TAx$ where $A = (a_{ij})$ be the $n×n$ matrix over $K$ whose entries are the coefficients of $q$.
Whenever I saw a matrix I imagine some kind of transformation. And it always happen with right multiplication. I never saw any transformation with left multiplication even I don't know is it really meaningful or not$?$ Can you provide an example of a transformation with left multiplication
$\color{red}{\text{Why this similarity$(*^TA* )$ pop up in these forms}?}$ Eventually these matrices are quite interesting. Because their properties or axioms which they have to follow are also reflect in those matrices.
Question $2$: Binary quadratic forms are of the form $ax^2+bxy+cy^2$ represented by the symmetric matrix $$\begin{pmatrix}a & b\\c & d\end{pmatrix}$$ I knew it from wikipedia. But I am curious about the general equation of the second degree $ax^2+bxy+cy^2+dx+ey+f=0$. Can $ax^2+bxy+cy^2+dx+ey+f$ also be written as a matrix$?$ Because if it $\color{red}{\text{is possible}}$ than I have an idea to make it diagonalization which will give me the standard form of that general equation.
Thanks for your time. Thanks in advance .