View Single Post
Screwdriver
#1
Jan19-11, 09:11 PM
P: 123
1. The problem statement, all variables and given/known data

Not really a problem per se; more of an issue with some aspects of linear transformations. We've learned that a linear combination of linear transformations is defined as follows:

[tex](c_1T_1+c_2T_2)(\vec{x})=c_1T_1(\vec{x})+c_2T_2(\vec{x})\,\,\,\,\,\, \vec{x}\varepsilon \mathbb{R}^n\,\,\,c\varepsilon \mathbb{R}[/tex]

And if T1, T2 are linear, then so is c1T1 + c2T2 and

[tex][c_1T_1+c_2T_2]=c_1[T_1]+c_2[T_2][/tex]

Where [T] is the standard matrix of T and c is some constant.

The problem is, is that I don't know why T1 and T2 being linear implies that c1T1 + c2T2 is linear other than by noting that multiplying the transformations by c and adding them together is a linear combination, but that doesn't seem to be a very good proof. The second part there would seem to follow from this though just based on the fact that transforming some vector x is the same as multiplying it by the standard matrix of T.

Also, linear transformations don't make much sense in general. I was told that they're basically the same things as functions, but then all of a sudden we're adding them together and multiplying them left and right which never happened with f(x). Does that matter? Can I just think of those Ts as fs? If so, I think that will help a lot because, as it is, the T seems very strange and foreign to me.

I would appreciate any tips on such matters
Phys.Org News Partner Science news on Phys.org
'Office life' of bacteria may be their weak spot
Lunar explorers will walk at higher speeds than thought
Philips introduces BlueTouch, PulseRelief control for pain relief