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 T:R→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 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
