I have a very stupid and simple question, which I do not have a clear idea on. One article (Third Paragraph of Page-199) that I read said,
"As a basic relationship in linear algebra states, the scalar product of vectors $x$ and $y$ in a base space of $A$ is $xAy$."
The snapshot of corresponding page.
What does the 'base space' specifically represent? Is there any other term that indicates the same concept? What is the difference between ordinary scalar or inner product $xy$ and $xAy$? What role base space A does play? What are the impacts by converting $xy$ into $xAy$? Thank you so much in advance!
In this case, we're talking about a finite dimensional vector space $V$ over a field $F$, equipped with some bilinear form $V\times V\to F$. One can say that the set $V$ is the base space for the bilinear form. You can also call it the "underlying space."
Then, when generating a geometric(/Clifford) algebra using $V$ and this bilinear form, you still can refer to $V$ as the "base" for the algebra since it is the key thing you need to generate the algebra.
$V$ is important it is usually used to model geometric things very directly, whereas elements of the Clifford algebra are interpreted as operating on the things in the underlying space. For example rotors are elements of the algebra that act on an image of $V$ in the Clifford algebra, modeling rotation in $V$.
Really, there are a couple of things to get your head around here. The first one is the idea of a bilinear form $B:V\times V\to F$. This is an abstract thing with particular properties.
If you fix a basis of $V$ then you can express a bilinear form as $B(x,y)=x^\top Ay$ where $x,y$ are column vectors and $A$ is a matrix, and the second half of the equality is matrix multiplication. One special case of this is when $A$ is the identity matrix, and then you have the regular inner product.
It's important to realize that $A$ depends on the basis you chose. $A$ is not unique to $B$, $A$ is unique with respect to $B$ and the basis. If you transform from one basis to another using a matrix $X$, then $A$ changes to $X^\top A X$. Under some circumstances, you may be able to take a matrix $A$ and find a change of basis so that $X^\top AX=I_n$ in the new basis, and then your new basis makes your bilinear form look like the dot product.
Another thing to realize is that if $B$ is symmetric (that is, $B(x,y)=B(y,x)$ for all $x,y$) then $A$ is a symmetric matrix.
I am not really sure if you actually mean $xAy$, since normally we like to interpret this as matrix multiplication $x^\top Ay$. However, it is totally understandable why one might write $xBy$ instead of $B(x,y)$ for aesthetic reasons.
P.S.: I could not access the article you're talking about.