Example comparing bilinear to linear maps

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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 [ilmath]T:\mathbb{R}\rightarrow\mathbb{R} [/ilmath] with [math]T(x)=x+x[/math]
is a linear map

To be a linear map [math]T(ax+by)=aT(x)+bT(y)[/math], so take:

[math]T(ax+by)=ax+by+ax+by=a(x+x)+b(y+y)=aT(x)+bT(y)[/math] as required.

Given the field was [ilmath]\mathbb{R} [/ilmath] we could have used the number [math]2[/math] of course. However this proof works for any field.

Thus addition is a linear map.

Addition is not bilinear

TODO: easy