Help in question, Inner product space.

Question

given those vectors u=<a,b> and v=<c,d> (above R)

<u,v> defined for real M -> ac-3ad-3bc+Mbd.

i need to find all M valuves that <u,v> defines an inner product

ive tried to use the conditions : <u,v> = <v,u>

<u,u> >= 0, and <u,u> = 0 if and only if u = 0,

<u+v,w> = <u,w> + <v,w>

and yet i struggle to find any M. can anyone help me please to figure this out?

Answer 1

Hint: Your function $\langle \cdot, \cdot \rangle$ can be written as $$ \left\langle \pmatrix{a\\b}, \pmatrix{c\\d} \right\rangle = \pmatrix{a&b}\pmatrix{1&-3\\-3&M} \pmatrix{c\\d}. $$ For every $M$, this function satisfies all of requirements except possible for the requirement that $\langle u,u \rangle > 0$ for all non-zero $u \in \Bbb R^2$.

For an approach without linear algebra:

Note that $$\langle u,u \rangle = (a-3b)^2 + k b^2$$ for some $k$ that depends of $M$.