(a) Let $u = (u_1, u_2)$ and v = (v1, v2). Prove that $\langle u, v \rangle = 3u_1v_1 + 5u_2v_2$ defines an inner product on R2 by showing that the inner product axioms hold.

(b) What conditions must k1 and k2 satisfy for = k1u1v1 + k2u2v2 to define an inner product on R2?

For part a, how do I prove that the inner product axioms hold with the coefficient values of 3 and 5?

For part b, how would I find the values k1 and k2, when I don't know the initial vector values of ?

  • 1
    $\begingroup$ Just start going through the axioms one by one. $\endgroup$ – user137731 Dec 1 '15 at 22:22
  • $\begingroup$ On an unrelated note, using MathJax to format your questions/ answers makes them much more readable. I've edited a couple of your equations to show you how it's done. You should try to finish editing it yourself. $\endgroup$ – user137731 Dec 1 '15 at 22:24

You have, by definition, $$\langle (u_1,u_2), (v_1, v_2)\rangle = 3u_1v_1 + 5u_2v_2$$

Now, you need to prove the following properties of inner products:

  • You need to prove that $$\langle (u_1,u_2), (v_1, v_2)\rangle = \langle(v_1,v_2),(u_1,u_2)\rangle$$
  • You need to prove that if $\alpha\in\mathbb R$, then $$\langle \alpha (u_1,u_2), (v_1, v_2)\rangle=\alpha\langle (u_1,u_2), (v_1, v_2)\rangle$$
  • You need to prove that $$\langle(u_1, u_2) + (w_1,w_2), (v_1,v_2)\rangle = \langle(u_1,u_2), (v_1,v_2)\rangle+\langle(w_1,w_2) , (v_1,v_2)\rangle$$
  • You need to prove that if $(u_1,u_2)\neq (0,0)$, then $$\langle(u_1,u_2),(u_1,u_2)\rangle>0$$ and that it is equal to $0$ if $(u_1,u_2)=(0,0)$

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