Let A be a fixed $n \times n$ nonsingular matrix. We define $\langle\mathbf u,\mathbf v \rangle = (A\mathbf u) \cdot (A\mathbf v)$

Prove that ⟨u,v⟩ is an inner product on $R^n$.

In class we let $\mathbf v=(x_1,\ldots,x_n)$ and proved

$$\langle \mathbf v,\mathbf v \rangle= x_1 \overline{x}_1 + x_2 \overline{x}_2 + \cdots + x_n \overline{x}_n =|x_1|^2+\cdots+|x_n|^2\ge0$$

It was also shown that an inner product can be expressed with matrix multiplication as uv=u$^t$v

So my question is how can I "inject" (for lack of a better word) ⟨u,v⟩=(Au)∙(Av) into the above proof?

The part that I'm having a hard time with is Au and Av, so v=$(x_1,..,x_n)$ but how do I write Av? is it just Av=A$(x_1,..,x_n)$?

If so, do I simply write ⟨v,v⟩=(Av)∙(Av)=A(${x_1}\overline{x_1}$+${x_2}\overline{x_2}$+...+ ${x_n}\overline{x_n}$)=A($|x_1|^2+...+|x_n|^2$)$≥0$?

Any help would be greatly appreciate.

Thank you,


Let $w=Av$

$$\langle v, v \rangle = (Av)^* (Av) = w^*w = \sum_{i=1}^n |w_i| \ge 0$$

Remark about your attempt: We can't just drop a copy of $A$ without justification.


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