How Does the Density of Rational Numbers Prove Matrix Linearity?

[bobbarker]

Homework Statement

Show that if F is continuous on Rⁿ and F(X+Y) = F(X) + F(Y) for all X and Y in Rⁿ, then A is linear. Hint: Rational numbers are dense in the reals.

Homework Equations

A transformation A is linear iff A(X) = (a matrix)
[ a₁₁x₁+...+a_1nx_n ]
[... ... ...]
[ a_m1x₁+...+a_mnx_n ]

The Attempt at a Solution

F(X) = A(X) is continuous and F(X+Y) = A(X+Y) = F(X) + F(Y) = A(X)+A(Y)

I feel like this basically proves itself...since a tranformation A is linear if A(X+Y) = A(X) + A(Y)... I don't understand where the denseness of rational numbers comes in?

Any help is greatly appreciated! :)

 

[tiny-tim]

hi bobbarker!

… a tranformation A is linear if A(X+Y) = A(X) + A(Y)

nooo …

a transformation A is linear if A(aX) = aA(X) for any scalar a ​