Linear Functions: Showing Properties for Multi-Vars

[WittyName]

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

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.