Proving that the dot product $\mathbb{R}^n\times \mathbb{R}^n\mapsto\mathbb{R}$ is bilinear

Question

My lecturer presented a proof of the fact that the dot product $\mathbb{R}^n\times \mathbb{R}^n\mapsto\mathbb{R}$ is bilinear. To show that the map is linear with $y$ fixed he wrote the following: $$\langle\lambda u+v, y\rangle=(\lambda u+v)^{t} y=\left(\lambda u^{t}+v^{t}\right) y=\lambda u^{t} y+v^{t} y=\lambda\langle u, y\rangle+\langle v, y\rangle$$

But isn't he using the linearity of $y$ in this proof? How does that make sense when it is the linearity itself he is trying to prove?

Answer 1

Your lecturer used the definition of the dot product: if $a=(a_,...,a_n), b=(b_,...,b_n), c=(c_,...,c_n) \in \mathbb{R}^n$ then by definition $\langle a+b, c\rangle= \sum_i ^n(a_i+b_i) \cdot c_i= \sum_i a_i c_i+ \sum_i b_i c_i$ since the sum lives in $\mathbb{R}$