Linear Functions: Showing Properties for Multi-Vars

[WittyName]

Linear functions are functions which satisfy the two properties:
$f(x+y)=f(x)+f(y) \\ f(a*x)=a*f(x)$

I was wondering how would you show this property was true for multi-variable functions e.g. $f(x,y,z).$ Would it suffice to show $$f(x_{1}+x_{2},y,z)=f(x_1,y,z)+f(x_2,y,z) \\ f(a*x,y,z)=a*f(x,y,z)?$$ Basically fix all other variables and show the properties are true for one variable, then repeat for the next variable different to the one we chose before.

Or would you have to consider something like $$f(x_{1}+x_{2},y_{1}+y_{2},z_1+z_2) \ \text{and} \ f(a*x,a*y,a*z)?$$

 

[chiro]

Hey WittyName and welcome to the forums.

Basically for multi-variable functions, your function is a matrix applied to a vector and if you can show that such a matrix exists that defines your function, then it's essentially linear.

what you basically do is treat your x as a vector (typically a column vector) and then show that a matrix exists to define your function.

The linearity works because of the nature of matrix multiplication and the properties of multiplying matrices by scalars as well as the distributivity of addition with multiplication where (X+Y)Z = XZ + YZ if all of these are matrices and have the right definitions (i.e. dimension wise).

 

[WittyName]

Hey WittyName and welcome to the forums.

Basically for multi-variable functions, your function is a matrix applied to a vector and if you can show that such a matrix exists that defines your function, then it's essentially linear.

what you basically do is treat your x as a vector (typically a column vector) and then show that a matrix exists to define your function.

The linearity works because of the nature of matrix multiplication and the properties of multiplying matrices by scalars as well as the distributivity of addition with multiplication where (X+Y)Z = XZ + YZ if all of these are matrices and have the right definitions (i.e. dimension wise).

Thanks for the reply.

Would you know of any sites where I can read more on this?

 

[chiro]

Here is the basics:

http://en.wikipedia.org/wiki/Linear_function

For vector spaces (again the basics):

http://mathworld.wolfram.com/VectorSpace.html

You could also look around for suggested books on linear algebra.

 

[blue_raver22]

I would explain that the properties of linear functions hold true for multi-variable functions as well. In fact, the two properties mentioned above are known as the linearity properties and they apply to all linear functions, regardless of the number of variables present. Therefore, it would suffice to show that these properties hold true for one variable at a time, while keeping the other variables fixed.

For instance, to show that f(x_{1}+x_{2},y,z)=f(x_1,y,z)+f(x_2,y,z), we can fix y and z, and vary x_{1} and x_{2}. If the equation holds true for all values of x_{1} and x_{2}, then the property is satisfied for the variable x. Similarly, we can repeat this process for the other variables to show that the linearity properties hold true for all variables in the function.

Alternatively, we can also consider the case of multiple variables at once, such as f(x_{1}+x_{2},y_{1}+y_{2},z_1+z_2). In this case, we can expand the function using the linearity properties and show that it simplifies to the same expression. This would also prove that the properties hold true for multi-variable functions.

In conclusion, the linearity properties apply to all linear functions, regardless of the number of variables present. To show that these properties hold true for multi-variable functions, we can either fix all variables except one and vary that one, or consider the case of multiple variables at once and show that the properties still hold true.