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What is a nonconstant linear function? 
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#1
Apr412, 11:11 AM

P: 5

arent linear functions always constant?



#2
Apr412, 11:21 AM

P: 836

No, they aren't. Concider for example f(x)=x.



#3
Apr412, 11:19 PM

Sci Advisor
P: 1,169

Actually, the only constant function that is linear is the 0 function.
If you have a linear map T:V>W between V.Spaces (this generalizes to rings, etc.) then, if T(v)==wo , i.e., T(v)=wo for all v in V, then: T(v+v')=wo≠ T(v)+T(v')=wo+wo=2wo. A similar argument applies to maps from a vector space to its base field. 


#4
Apr512, 03:41 PM

P: 5

What is a nonconstant linear function?



#5
Apr512, 03:48 PM

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PF Gold
P: 4,500

A constant function is a function which always takes the same value, for example f(x)=2.
All linear functions on R^{n}can be written as y=Ax where A is a matrix (in one dimension, just a number) 


#6
Apr512, 04:05 PM

Sci Advisor
P: 1,169

spoke:
You may be confusing constant rate of change, i.e., constant derivativea property of linear functions with constant function. 


#7
Apr512, 10:45 PM

P: 5




#8
Apr612, 12:37 AM

Sci Advisor
P: 1,169

Yes, exactly, that is what a constant function is like when seen as a subset of AxB.
Not to nitpick, but you may want to specify the sets A,B where you are defining your function as a subset of AxB; here, A is clearly specified, but it is not clear what B is (unless you assume your function is onto B). 


#9
Apr612, 01:02 AM

P: 3,014

A linear function is constant if and only if its slope is zero. By contaposition, a linear function is not constant (i.e. nonconstant) iff its slope is different from zero.



#10
Apr612, 01:56 AM

Sci Advisor
P: 1,169

Your right, Dickfore, but your example is that of a map from ℝ to itself may be too
specific for a general definition of function. 


#11
Apr612, 09:46 AM

P: 3,014

[tex] \mathbf{y}_{n \times 1} = \hat{A}_{n \times m} \cdot \mathbf{x}_{m \times 1} + \mathbf{b}_{n \times 1} [/tex] This is a general mapping from [itex]\mathbb{C}^m \rightarrow \mathbb{C}^n[/itex]. But, now, the function may be constant in a more general case, when [itex]\mathrm{rank}A \le m < n[/itex]. 


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