Example comparing bilinear to linear maps

These examples are supposed to demonstrate some differences between linear maps and bilinear maps

Addition is a linear map

Here we will show that addition, given by:
Take $T:\mathbb{R}\rightarrow\mathbb{R}$ with

$$T(x)=x+x$$
is a linear map

To be a linear map

$$T(ax+by)=aT(x)+bT(y)$$ , so take:

$$T(ax+by)=ax+by+ax+by=a(x+x)+b(y+y)=aT(x)+bT(y)$$ as required.

Given the field was $\mathbb{R}$ we could have used the number

$$2$$ of course. However this proof works for any field.

Thus addition is a linear map.

Addition is not bilinear

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TODO: easy

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