I am reading a linear algebra book.
The typical definition of an inner product is the following:
Let $F=\mathbb{R}$ or $\mathbb{C}$.
Let $V$ be a vector space over $F$. An inner product on $V$ is a function that assigns, to every ordered pair of vectors $x$ and $y$ in $V$, a scalar in $F$, denoted $\langle x,y\rangle$, such that for all $x,y$, and $z$ in $V$ and all $c$ in $F$, the following hold:
(a) $\langle x+z,y\rangle=\langle x,y\rangle+\langle z,y\rangle.$
(b) $\langle cx,y\rangle=c\langle x,y\rangle$.
(c) $\overline{\langle x,y\rangle}=\langle y,x\rangle$, where the bar denotes complex conjugation.
(d) If $x\neq 0$, then $\langle x, x\rangle$ is a positive real number.
Linear algebra books don't adopt the following definition for an inner product.
Why?
Isn't this definition ok?
I like this definition.
Let $F=\mathbb{R}$ or $\mathbb{C}$.
Let $V$ be an $n$-dimensional vector space over $F$.
Let $B=\{v_1,\dots, v_n\}$ be a basis of $V$.
We define a function $f_B$ from $V\times V$ to $F$ by $f_B(v,w):=x_1\overline{y_1}+\dots+x_n\overline{y_n}$, where $v=x_1v_1+\dots+x_nv_n$ and $w=y_1v_1+\dots+y_nv_n$.
$g$ is called an inner product on $V$ if $g=f_B$ for some basis $B$ of $V$.