For example the inner product where $u=(u_1,u_2) , v=(v_1,v_2)$ is $\left<u,v\right>= 3u_1v_2 + u_2v_1$ How would you prove that it is not symmetric.
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The function $\langle u,v\rangle=3u_1v_2+u_2v_1$ can be shown not to be symmetric by simply plugging in some values. For example, taking $(1,0)$ and $(0,1)$ it is clear that $$\langle(1,0),(0,1)\rangle\neq\langle(0,1),(1,0)\rangle.$$ Do note that this function does not define an inner product; it is not (conjugate) symmetric, and it is not positive definite.