I'm a little stuck with some concepts and don't find how to unblock. Possibly it's not a clever question but I need the help of someone to unblock. The problem is related to the concept of "base" and "cartesian coordinates system" in $R^n$.
First example:
Suppose the base $B$ = $\{e_1, e_2\}: Gen\{B\} = R^2$. A vector x expressed in this base is denoted $[x]_b$. Let the matrix $A=[e_1 \space \space e_2]$. The multiplication A $[x]_b$ return the vector $x$ which is expressed in the standard base. So it means that the base $B$ was defined respect the standard base and also when we talk about calculating the vector $x$ we are referring implicitly to a standard base. Why?
Second example
Second: Suppose that the base $B$ is not orthogonal (oblique). As we said before, the coordinates of the vectors $[x]_b$ are referred to the standard base, so if we calculate the dot product: $<e_1, e_2> \neq 0$. But what would it happen if we calculate this product with the vectors $\{e_1,e_2\}$ expressed from base B (itself). In this case: $e_1 = (1,0)$ and $e_2 = (0,1)$ so $<e_1, e_2> = 0$ and it does not make sense because they are not orthogonal. Where is the mistake? Are a base and cartesian coordinates the same concept?
I know that these are a little bit and strange questions but I don't know how to unblock myself and any books that I review does not talk about these topics. Apart of these question, I would be grateful if someone recommends me a book of linear algebra that deep in these topics. I look for get more insights about this topic.
A lot of thanks in advance.
When dealing with linear functions, there are two concepts which are interchangeable (at least for finite vector spaces). I call this concepts the "world of vectors" and the "world of tuples".
Strictly speaking, vectors and tuples are not the same: Let $V$ be a vector space over the field $K$. Then the elements of $V$ are called vectors, whereas the elements of $K$ are called scalars. The elements of $K^n$ are called $n$-tuples.
Tuples and vectors are related to each other via the concepts of bases and coordinates. As a matter of fact, tuples are often called vectors, which makes it hard for beginners the notice the difference between vectors and tuples.
You cannot really multiply a matrix with a vector: Strictly speaking, you multiply a matrix with a tuple and the result of the operation is also a tuple.
So let's get into details. Let $x$ be a vector (an element of $V$). A priori, $x$ doesn't have coordinates. Coordinates are always relative to a basis. So let $B=(b_1,\ldots,b_n)$ be a basis of $V$. (Please note: For simplicity of notation, I assumed that $V$ has finite dimension.) Then you can write $x$ as $x=\sum_{i=1}^n a_i b_i$ (with $a_i\in K,b_i\in B$). The $a_i$ are called the coordinates of $x$ with respect to $B$.
Now let $K$ be a field, $V,W$ be vector spaces over $K$ with $\dim(V)=n$ and $\dim(W)=m$, $f:V\to W$ be a linear function and $v\in V$. By applying $f$ to $v$, you get $w=f(v)$ with $w\in W$. This is the calculation in the "world of vectors". In the "world of tuples", you additionally need bases to express this relationship: Let $B_V$ be a basis of $V$ and $B_W$ be a basis of $W$. If $[v]_{B_V}$ denotes the $n$-tuple that represents the coordinates of $v$ with respect to $B_V$, $[w]_{B_W}$ denotes the $m$-tuple that represents the coordinates of $w$ with respect to $B_W$, and $A\in K^{m\times n}$ denotes the matrix which represents the linear function $f$ with respect to the bases $B_V$ and $B_W$, you can write $[w]_{B_W} = A [v]_{B_V}$ instead of $w=f(v)$.
$w=f(v)$ is a form of calculating with vectors, whereas $[w]_{B_W} = A [v]_{B_V}$ is a form of calculation with tuples. But linear algebra tells us that both ways of calculating are interchangeable (at least for finite vector spaces).
As for your second example: Speaking in the "world of vectors", $\langle .,.\rangle$ is a bilinear form. Two vectors $x,y$ are called orthogonal iff $\langle x,y\rangle=0$. You can perform this calculation in the "world of tuples", but you have to not only represent $x$ and $y$ by coordinates with respect to some bases, but you must also represent $\langle .,.\rangle$ by its coordinates with respect to the sames bases. And this is where your mistake lies: You neglected the fact that $\langle .,.\rangle$ has to be translated into the "world of tuples" as well.