Difference between revisions of "Example comparing bilinear to linear maps"

Latest revision as of 16:30, 23 August 2015

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

TODO: easy