I have to Show, if a function $f(u,v)=1+3x_{1}-2y_{2}$ is Bilinear. $u^T =(x_{1},x_{2}) v^T =(y_{1},y_{2}) $ I know, that I have to prove
$ ⟨ v , w 1 + w 2 ⟩ = ⟨ v , w 1 ⟩ + ⟨ v , w 2 ⟩ \\ {\displaystyle \langle v,w_{1}+w_{2}\rangle =\langle v,w_{1}\rangle +\langle v,w_{2}\rangle } ,\\ ⟨ λ v , w ⟩ = λ ⟨ v , w ⟩ {\displaystyle \langle \lambda v,w\rangle =\lambda \langle v,w\rangle } ,\\ ⟨ v , w λ ⟩ = ⟨ v , w ⟩ λ {\displaystyle \langle v,w\lambda \rangle =\langle v,w\rangle \lambda } .$ \
How can I do it? Thanks.
The function $f$ is not bilinear. Why? Take $u^T=(1,1)$, ${v_1}^{T}=(1,1)$ and ${v_2}^{T}=(1,1)$, then
$$f(u,v_1+v_2)=f((1,1),(2,2))=1+3(1)-2(2)=0$$ and on the other hand,
$$f(u,v_1)+f(u,v_2)=f((1,1),(1,1))+f((1,1),(1,1))=2+2=4$$