How could a bilinear form with positive coefficients fail to be an inner product?

Question

So to prove somethings an inner product it has to be all positive terms. But why do not all such expressions with positive coefficients define inner products?

For example, how is $$\left<u,v\right> = x_1y_1 + 2x_1y_2 + x_2y_1 + 3x_2y_2$$ not an inner product?

Answer 1

$$\langle (1,0), (1,0)\rangle = 0, \qquad (1,0) \neq {\bf 0}.$$

It could be degenerate, like the above example shows.

Answer 2

If you drop the requirement of positive-definiteness, you can define many scalar products $\langle\mathbf u,\mathbf v\rangle=(\mathbf v,A\mathbf u)$, where $A$ is a symmetric matrix and $(\cdot,\cdot)$ is the Euclidean scalar product. These scalar products are closely related to quadratic forms.