- #1
bobbarker
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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+...+a1nxn ]
[... ... ...]
[ am1x1+...+amnxn ]
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! :)