What can we do with a basis that we can't do without it?

Question

All my life, I've learned to treat vectors as a set of 3 real numbers, which I multiply by a basis to get the actual vector: $\langle a, b, c\rangle = ae_1+be_2+ce_3$. As a programmer this is convenient because we can notate three real numbers in memory.

Recently, I've been looking at vectors and their kin from a different perspective. Like the classical way of viewing tensors, the vector is simply itself. There are operations I can do on it as a vector(like a dot product, or scalar product). These operations work perfectly without a basis. Its like everything I learned with a basis didn't need one at all!

What can I do with a vector if I have a basis that I can't do without one? The only operation I have found is "map a vector into 3 real numbers," which is useful to me as a programmer to encode a vector, but other than that, I'm having trouble finding what operations cannot be done without a basis.

Answer 1

You are implicitly using bases whenever you are performing some operation like dot product, cross product, scaling etc. on vectors.

The dot product is simply the sum of the products of coefficients for each base. You may not write the bases, but that doesn't mean you aren't using them. All operations deal with the coefficients of the bases.

EDIT: The application for bases is much more evident when you need to use alternate bases.

Answer 2

One example: with a fixed basis, you can find a canonical isomorphism from your vector space $V$ to its dual $V^*$ of linear functionals (I'm assuming finite dimensions here).

For a given basis $(v_1, v_2, \ldots, v_n)$ we can define a linear functional $f_i$ by defining its action on the basis, in particular, by defining $f_i(v_i) = 1$ and $f_i(v_j) = 0$ for $j \neq i$. In this way, we define the dual basis $(f_1, \ldots, f_n)$ for $V^*$ of $(v_1, \ldots, v_n)$.

This gives us a canonical isomorphism from $V$ to $V^*$, where we map $$a_1 v_1 + \ldots + a_n v_n \mapsto a_1 f_1 + \ldots + a_n f_n.$$ (Such a map is well-defined, since $(v_1, \ldots, v_n)$ is a basis.)

When you don't have a basis specified, there will obviously still be many such isomorphisms (e.g. pick any basis you want, and form an isomorphism as above), but there won't be an obvious, canonical isomorphism between the spaces.

Answer 3

Some operations like scalar product, cross product or general linear maps are independent of the choice of basis. There are some operations that are not independent. A simple example is given by the projection to a basis-vector.

If $v=v_1\cdot e_1 +v_2 \cdot e_2$ where $v_1,v_2 \in \mathbb{R}$, then the map $\mathbb{R}^2 \to \mathbb{R}, v \mapsto v_1$ depends on the basis $\{e_1,e_2\}$.