View source for Example comparing bilinear to linear maps

These examples are supposed to demonstrate some differences between [[Linear map|linear maps]] and [[Bilinear map|bilinear maps]]

==Addition is a linear map==
Here we will show that addition, given by:<br/>
Take {{M|T:\mathbb{R}\rightarrow\mathbb{R} }} with <math>T(x)=x+x</math><br />
is a [[Linear map|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|field]] was {{M|\mathbb{R} }} 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}}
{{Example|Linear Algebra}}