Difference between revisions of "Example comparing bilinear to linear maps"
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==Addition is a linear map== | ==Addition is a linear map== | ||
| − | Here we will show that addition | + | Here we will show that addition, given by:<br/> |
| − | Take {{M|T:\mathbb{R}\rightarrow\mathbb{R} }} with <math>T(x)= | + | Take {{M|T:\mathbb{R}\rightarrow\mathbb{R} }} with <math>T(x)=x+x</math><br /> |
is a [[Linear map|linear map]] | is a [[Linear map|linear map]] | ||
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<math>T(ax+by)=ax+by+ax+by=a(x+x)+b(y+y)=aT(x)+bT(y)</math> as required. | <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 | + | 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. | Thus addition is a linear map. | ||
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
