Proving a sphere is made up of points of certain radius(?) 
Let $S_1$ be the sphere of radius $1$, centered at the origin. Let $a$ be a number $> 0$. If $X$ is a point of the sphere $S_1$, then $aX$ is a point of the sphere of radius a, because $\|aX\| = a\|X\| = a$. Prove we get all points of the sphere of radius $a$ in this manner.

Please, help me reword this problem and understand exactly what's being asked. I'd appreciate some hints, too.  
Thanks.
 A: Many questions about mappings involve surjections, injections, or bijections. Often to solve a problem we have to figure out what it's asking for us to prove. In general, if $A$ is the input set, $B$ the output set, and $f$ the mapping with domain $D_f=A$, then:


*

*If we are asked to show every element $b \in B$ is a possible value of $f(a)$ for at least one $a \in A$, then we must show surjection from $A \to B$.

*If we are asked to show that every element $a \in A$ maps to a unique element $b \in B$ (that is, there is no $a' \ne a$ such that $f(a') = b$), then we must show injection from $A \to B$.

*If we are asked to show both surjection (at least one input for every output) and injection (at most one input for every output), then we must show bijection. Bijection is shown if we can create an inverse function $f^{-1}: B \to A$.


This question can be reworded as:

Prove that $f: X \to aX$ is a surjection from $S_1$ to $S_a$ for $a>0$.

Hint: As noted in the question, note that $f(X)=rX$ has distance $r$ from the origin; hence, $f(X) \in S_r$. This can be generalized to the following lemma:

Given $X \in S_c$, then $f(X)=rX \in S_{c \cdot r}$.

A: So for any vector $\mathbf{x}$ in the unit sphere, we can scale it by a factor of $a$ to get a new point $a\mathbf{x}$ which is in the sphere of radius $a$.
The question is asking if all points on the sphere of radius $a$ can be expressed likewise - as scaled versions of vectors in the unit sphere.
The answer would seem to be yes: for $\mathbf{y}$ in the sphere of radius $a$, note that $$\mathbf{y}=a\left({1\over a}\mathbf{y}\right)$$
