Difference between revisions of "Example comparing bilinear to linear maps"
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| + | These examples are supposed to demonstrate some differences between [[Linear map|linear maps]] and [[Bilinear map|bilinear maps]] | ||
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==Addition is a linear map== | ==Addition is a linear map== | ||
Here we will show that addition, given by:<br/> | Here we will show that addition, given by:<br/> | ||
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 [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
