let $T_A:\mathbb R^n\to \mathbb R^n$, and $A$ is vector in $\mathbb R^n$ such that $$T_A(X)=X+A$$$T_A$ is called the translation by $A$, but I don't understand why we call this mapping a translation, because if you add a vector $X$ to another vector $A$ you will get a new vector, not the translation of $A$, if $X+A$ is the translation of $X$ then we get that $$\mid \mid X+A\mid \mid = \mid \mid X \mid \mid$$Which is not true unless $A=O$.
In this picture, let $A=(1,1)$ and $X=(5,2)$, and you can see the diffrence between adding the vector $X+A=C$ and translating $X$ by $A$
2 Answers
$\begingroup$You are confusing two ways of thinking about a "vector". One is that it's an arrow starting at the origin. The other is that it's just an $n$-tuple of coordinates - that is, a point in $n$-space.
The map you describe translates points in space, each the same way determined by $A$. In your picture that's diagonally up and right by a distance $\sqrt{2}$.
It's often useful (particularly in physics) to think of a vector as an arrow. It's just not right in this context.
$\endgroup$ 2 $\begingroup$Think of $X$ as a point in space. Then $X + A$ is precisely the result of translating $X$ by $A$ (thought of as a vector in the sense that you are thinking of them).
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