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Let $T\colon\mathbb{R}^2\to\mathbb{R}^2$ be given by $T(x,y) = (x + 2y, 3x + y)$.

I'm not sure if I plugged the values in the right place.

Ax. 1: $T(x + y) = T(x) + T(y)$

$[(x+2y) + 2(3x + y) + 3(x + 2y) - (3x - y)] = [(x+2y) + 2(3x + y)] + [3(x + 2y) - (3x - y)]$

Is that right?

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    That's not right - what you've computed on the left hand side is T(T(x,y)). You should probably use different variables for your first axiom, because T is a function of either two variables or a vector, but you're treating $x$ and $y$ there like they were real numbers; the $x$ and $y$ of the axiom aren't the same as the $x$ and $y$ of T's defining equation, and I think you may be confusing them. I'd suggest writing the axiom as $T(\vec{a}+\vec{b}) = T(\vec{a})+T(\vec{b})$ and then use $a_x$, $a_y$ and $b_x$, $b_y$ as the two components of $\vec{a}$ and $\vec{b}$.2011-04-15

3 Answers 3