Question

Suppose u; v is the element of V . Prove that || au + bv || = || bu + av || for all a, b is the element of R if and only if || u || = || v || .

Answer 1

Suppose that $V={\mathbb{R}}^n$. Then, we have: $u=(u_1,...u_n)$ and $v=(v_1,...,v_n)$. We have: $(au+bv)=(au_1+bv_1,...,au_n+bv_n)$ and $(bu+av)=(bu_1+av_1,...,bu_n+av_n)$. We must obtain the equality: $\sqrt{(au_1+bv_1)^2+...+(au_n+bv_n)^2}=\sqrt{(bu_1+av_1)^2+...+(bu_n+av_n)^2}$. Simplify the left side: $\sqrt{(a^2u^2_1+2abu_1v_1+b^2v^2_1+...+a^2u^2_n+2abu_nv_n+b^2v^2_n}.$ Right side has the form: $\sqrt{(bu_1+av_1)^2+...+(bu_n+av_n)^2}=\sqrt{b^2u^2_1+2abu_1v_1+a^2v_1^2+...+b^2u^2_n+2abu_nv_n+a^2v_n^2}$

Since by assumption we have: $\sqrt{u_1^2+...+u_n^2}=\sqrt{v_1^2+...+v_n^2}$ . Using the latter we observe that left and right side are equal. Thus, the equality $||au+bv||=||bu+av||$ holds. We showed that if $||u||=||v||$, then the equality $||au+bv||=||bu+av||$ holds. Otherwise, we can set $a=1,b=0$ and obtain $||u||=||v||.$