Find a basis for the set of vectors of magnitude $1$

Question

Is it possible to find a basis for the set of vectors that have a magnitude of 1?

I've studied a bit of linear algebra in school, and if the question can be answered with linear algebra, I can't think of it. Would appreciate some help.

Attempt

Let $S$ be the set of vectors $\begin{bmatrix} x_1 \\ x_2 \end{bmatrix}$ such that $x_1^2 + x_2^2 = 1$

I thought of writing $x_1$ and $x_2$ in terms of an angle $\theta$, but I don't know where to go from there either.

Answer 1

As the other answer points out, it doesn't really make sense to ask for a "basis" (in the sense of vector spaces) for something that isn't a vector space. There isn't a particularly deep reason that this is true, it's just due to how vector spaces and bases are defined: in particular, we only define a "basis of $V$" where $V$ is some vector space; that's it.

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It would be OK to ask if you can parameterize your set $S$ -- and indeed, you can. This is essentially how the longitude/latitude system for the Earth works; we specify a point on the Earth's surface by specifying two angles (one measuring North/South from the equator, another measuring East/West from some specified starting "line"). So you can use two parameters to specify a point in $S$ but doing so doesn't yield any sort of "basis" because $S$ isn't a vector space.

Essentially, the take away here is that linear algebra is not exactly meant to study spheres :)

Answer 2

The issue here is that the collection of all unit vectors in a normed linear space is not a subspace, and therefore cannot have a basis!

For example, $\langle1,0\rangle,\langle0,1\rangle\in\Bbb R^2$ but $$ \langle1,0\rangle+\langle0,1\rangle=\langle1,1\rangle $$ is not a unit vector.

Of course, if you wish to find a basis for a normed linear space consisting of unit vectors, simply start with any basis $\mathcal B$ and consider $\mathcal U=\{v/\lVert v\rVert:v\in\mathcal B\}$.