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What is a nonconstant linear function?

by spoke
Tags: function, linear, nonconstant
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spoke
#1
Apr4-12, 11:11 AM
P: 5
arent linear functions always constant?
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espen180
#2
Apr4-12, 11:21 AM
P: 836
No, they aren't. Concider for example f(x)=x.
Bacle2
#3
Apr4-12, 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.

spoke
#4
Apr5-12, 03:41 PM
P: 5
What is a nonconstant linear function?

Quote Quote by espen180 View Post
No, they aren't. Concider for example f(x)=x.
well then i dont know what a constant and nonconstant linear functions are. Because f(x)=x is linear when graphed, so i was assuming linear is synonymous the word constant. as in a constant rate of change or constant slope.
Office_Shredder
#5
Apr5-12, 03:48 PM
Emeritus
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P: 4,500
A constant function is a function which always takes the same value, for example f(x)=2.
All linear functions on Rncan be written as y=Ax where A is a matrix (in one dimension, just a number)
Bacle2
#6
Apr5-12, 04:05 PM
Sci Advisor
P: 1,169
spoke:

You may be confusing constant rate of change, i.e., constant derivative--a property of linear functions-- with constant function.
spoke
#7
Apr5-12, 10:45 PM
P: 5
Quote Quote by Office_Shredder View Post
A constant function is a function which always takes the same value, for example f(x)=2.
All linear functions on Rncan be written as y=Ax where A is a matrix (in one dimension, just a number)
So would this relation be an example constant function? {(1,2), (2,2), (3,2), (4,2)}
Bacle2
#8
Apr6-12, 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).
Dickfore
#9
Apr6-12, 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. non-constant) iff its slope is different from zero.
Bacle2
#10
Apr6-12, 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.
Dickfore
#11
Apr6-12, 09:46 AM
P: 3,014
Quote Quote by Bacle2 View Post
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.
OK, make
[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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