A Proof that f:R^n->R is linear if

In summary: R^n.In summary, we need to prove that a map f: R^n -> R is linear if f(0)=0 and f is differentiable, and tf(x)=f(tx) for all x in R^n. One approach is to take the derivative and use linearity, but since the line integral has not been defined in class, this may not be the intended method. Another approach is to show that the partials are all constant, which implies linearity. Finally, defining a function O(x,y)=f(x+y)-f(x)-f(y) and showing that O'(x,y)=0, we can prove additivity and thus linearlity of f.
  • #1
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Homework Statement



Prove that if a map f: R^n -> R if f(0)=0 satisfies f differentiable, tf(x)=f(tx) for all x in R^n, then f is linear. (Prove additivity)



Homework Equations





The Attempt at a Solution



So the first thing I tried was taking the derivative using the Matrix (or vector in this case)

We have that f'(x+y) = f'(x)+f'(y) by the linearity of the derivative.

Could one then take line integrals to get the desired equality (A path from 0 to x, a path from 0 to y), we should have, added together should enter a path from 0 to (x+y), so long as we are integrating from the same starting point (What's bothering me about this is it's not important f(0)=0, since we can just make the starting point of the path integral equal...)

However we haven't defined the line integral in class, so regardless I'm thinking this is probably not what the instructor wants us to gain from the problem. Anyone have any insights about how I might approach this (Other Ideas I have had include, Showing that the partials are all constant (If they are all constant, than the original function which produced them must have contained only linear dependence on each of the n co-ordinates... that would imply the original vector field was linear together with f(0)=0 yes?)

EDIT: So I explored the second Idea. If we differentiate (gradient) the second term we get that

tf'(tx)=tf'(x)-> f'(tx)=f'(x), so we know that if homogeneity holds, f'(x) is constant, which shows that all of the partials are constant. If all the partials are constant, then in each co-ordinate, we have linearity (partial f by partial x_1 = c, so that f at most depended only on x_1 in its first power). f is however a scalar field though, so while I have convinced myself of the truth of this and drawn pictures, how might I rigorously arrive at the equality I need (That f(x_i+y_i)=f(x_i)+f(y_i) for all i, where x_i denotes the projection of x onto the vector x_i,

EDIT: I think I got it.

Define a function O(x,y)=f(x+y)-f(x)-f(y)

Then O'(x,y)= f'(x+y)-f'(x)-f'(y), but since f'(x)=f'(y), f'(x+y)=f'(x)+f'(y) so that O'(x,y) = 0.

Then, since f(0)=0, O(0,0)=0, so that O(x) must be zero everywhere.
 
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  • #2
Prove that a map f: R^n -> R if f(0)=0 and f is differentiable. If in addition tf(x)=f(tx) for all x in R^n, prove f is linear. (Prove additivity)
Part of your first sentence is missing. Prove that a map f:R^n --> R is what? From the work you show, I could probably figure out what's missing, but I would rather you tell me than put in that effort.
 
  • #3
Sorry, we need to prove it's linear. Which since we have homogeneity, amounts to showing f(x+y)=f(x)+f(y)
 

1. What is a linear function?

A linear function is a mathematical function that maps a set of input values to a set of output values in a straight line. It has the form f(x) = mx + b, where m is the slope or rate of change and b is the y-intercept.

2. How is a linear function different from other types of functions?

A linear function differs from other types of functions in that it has a constant rate of change or slope. This means that for every unit increase in the input variable, the output value also increases or decreases by the same amount.

3. What does it mean for a function to be linear in n-dimensional space?

A function is considered linear in n-dimensional space if it satisfies the properties of a linear function in each of its n variables. This means that the function's rate of change or slope remains constant in each variable, and the function can be represented graphically as a straight line in n-dimensional space.

4. What is the significance of proving that a function is linear?

Proving that a function is linear is significant because it allows us to understand the behavior and properties of the function more deeply. It also enables us to make accurate predictions and solve complex problems using the function in various mathematical and scientific applications.

5. How can we prove that a function is linear in n-dimensional space?

We can prove that a function is linear in n-dimensional space by showing that it satisfies the two properties of linearity: additivity and homogeneity. Additivity means that the function's output for the sum of two input values is equal to the sum of the function's outputs for each individual input value. Homogeneity means that the function's output for the product of an input value and a scalar is equal to the product of the function's output for the input value and the scalar.

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