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Example of a linear function

An example of a linear function; We must show closure under scalar multiplication, and vector addition.

Define F by: F(X) = (X,Y)

F(X1+X2) = (X1+X2,Y) = (X1,Y) + (X2,Y)

= F(X1) + F(X2)

F(aX) = (aX,Y) = a (X,Y) = a F(X)

So F is linear, other examples include ordered tuples, and directed line segments.

Comments

I wtote my first blog using pointy brackets not parenthesis. If X and V are I dimensional vectors the function F is the Euclidian inner-product = x1*v1+x2*v2+...+xI*vI There are other properties associated with this multi linear function, but my point here is I wrtote F(X)= NOT (X,Y) a distinction that anyone who has worked the common notation of vector spaces will appreciate.