# List all inner product axioms that fail to hold

This is a question from the textbook Elementary Linear Algebra: Applications Version which I am really struggling with.

Let $$u = (u_1, u_2, u_3)$$ and $$v = (v_1, v_2, v_3)$$. Show that the following expressions does not define an inner product on $$\mathbb{R}^3$$, and list all inner product axioms that fail to hold.

a) $$\langle u, v\rangle = u_1^2v_1^2 + u_2^2v_2^2 + u_3^2v_3^2$$

a) $$\langle u, v\rangle = u_1v_1 - u_2v_2 + u_3v_3$$

I have been told that the solution for a) is that axioms 2) and 3) do not hold but I do not understand how to prove that. I don’t think I understand the premise of the question, how can $$\langle u, v\rangle = u_1^2v_1^2 + u_2^2v_2^2 + u_3^2v_3^2$$ if $$u = (u_1, u_2, u_3)$$ and $$v = (v_1, v_2, v_3)$$, wouldn’t it rather be $$\langle u, v\rangle = u_1v_1 + u_2v_2 + u_3v_3$$?

The Axioms listed in the text book are as follows:

1. $$\langle u, v\rangle = \langle v, u\rangle$$ (Symmetry axiom)
2. $$\langle u + v, w\rangle = \langle u, w\rangle + \langle v, w\rangle$$ (Additivity axiom)
3. $$\langle ku, v\rangle = k\langle u, v\rangle$$ (Homogeneity axiom)
4. $$\langle v, v\rangle \geq 0$$ and $$\langle v, v\rangle = 0$$ if and only if $$v = 0$$ (Positivity axiom)
• In each of these expressions, $\langle \cdot, \cdot \rangle$ denotes a new pairing (a function of two vectors that produces a scalar) that has nothing to do with the standard Euclidean inner product. Oct 31, 2022 at 6:12

When: $$\begin{equation*} u = (u_1, u_2, u_3) \end{equation*}$$ $$\begin{equation*} v = (v_1, v_2, v_3) \end{equation*}$$ $$\begin{equation*} \langle u, v\rangle = u_1^2v_1^2 + u_2^2v_2^2 + u_3^2v_3^2 \end{equation*}$$
LHS: $$\begin{equation*} \langle u + v, w\rangle = (u_1+v_1)^2w_1^2 + (u_2+v_2)^2w_2^2 + (u_3+v_3)^2w_3^2 \end{equation*}$$ $$\begin{equation*} \langle u + v, w\rangle = (u_1^2+2u_1v_1+v_1^2)w_1^2 + (u_2^2+2u_2v_2+v_2^2)w_2^2 + (u_3^2+2u_3v_3+v_3^2)w_3^2 \end{equation*}$$ $$\begin{equation*} \langle u + v, w\rangle = u_1^2w_1^2+2u_1v_1w_1^2+v_1^2w_1^2 + u_2^2w_2^2+2u_2v_2w_2^2+v_2^2w_2^2 + u_3^2w_3^2+2u_3v_3w_3^2+v_3^2w_3^2 \end{equation*}$$
RHS: $$\begin{equation*} \langle u, w\rangle + \langle v, w\rangle = u_1^2w_1^2+v_1^2w_1^2 + u_2^2w_2^2 + v_2^2w_2^2 + u_3^2w_3^2 + v_3^2w_3^2 \end{equation*}$$
$$\begin{equation*} \langle u, w\rangle + \langle v, w\rangle \neq \langle u + v, w\rangle \end{equation*}$$