Forms/Bilinearity/1/Exercise

Determine which of the following mappings ${}\varphi \colon \mathbb {R} ^{2}\times \mathbb {R} ^{2}\rightarrow \mathbb {R}$ are bilinear. In the positive case, determine whether the properties alternating and symmetric hold.

a) ${}\varphi (x,y):=x_{1}y_{1}$.

b) ${}\varphi (x,y):=x_{1}x_{2}+y_{1}y_{2}$.

c) ${}\varphi (x,y):=2x_{1}y_{2}+3x_{2}y_{1}$.

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