A few questions about dual bases and bilinear forms I'm on my journey to understanding linear algebra. I've reached the point where I have to understand the dual basis and bilinear forms. This is something I didn't find in the book that I learn from (David Poole's A Modern Introduction to Linear Algebra). My teacher at the university did some examples so I want to understand them.
I know what a basis is. I know also what the geometrical interpretation is. I also know what is a quadratic form (is it related in any way to bilinear forms?)
Could you please summarize what is a bilinear form and a dual base? Also, if there is any geometrical interpretation, please list it. Thank you a lot!

To answer just one of your questions, let me work over a field $k$ of characteristic $\neq 2$, such as $\mathbf R$. Then one can pass back and forth between quadratic forms and symmetric bilinear forms as follows. A quadratic form $f\colon k^n \to k$ on $k^n$ (thought of as column vectors) corresponds to a (unique) symmetric $n \times n$ matrix $A$ such that
$$
f(x) = {}^txAx. \qquad \text{[${}^tx$ is the transpose of the column vector $x$]}
$$
But we can also define a map $g\colon k^n \times k^n \to k$ by
$$
g(x, y) = {}^txAy,
$$
and this is a symmetric bilinear form (take the transpose of the right-hand side). Can you see how to go in the other direction?
