Why do we need “basis” in linear algebra?

I am very curios about why in linear algebra we need different basis, why can't we just have the standard basis and work with that? And how basis is used in computer graphics?

• Let $V$ be the subspace of $\Bbb R^2$ generated by $(1,2)$. What is "the standard basis for $V$? The unit vectors you were banking on are not available... – rschwieb Jun 7 '16 at 15:08
• The continuous functions on $[0,1]$ is a vector space under the usual operations of function addition and scaling by a constant. There's no standard basis. – DisintegratingByParts Jun 7 '16 at 15:09
• Let's say you are traveling on a long road heading northeast. It is convenient to measure your progress along the road rather than always using latitude-longitude coordinates, isn't it? Choosing a basis is the same as choosing your coordinate system, and often different coordinate systems are useful for different situations. – Jair Taylor Jun 7 '16 at 16:13
• It's a little bit like asking: why study automorphisms when you can just take the identity? – Justin Young Jun 7 '16 at 19:40

Sometimes the basis that is most convenient to use is different from the standard basis. For example, suppose $A$ is an $n \times n$ matrix which represents a linear transformation $T$. If $A$ has a set of eigenvectors which form a basis for the $n$-dimensional space, then with respect to this basis the linear transformation $T$ can be represented by a diagonal matrix. Diagonal matrices are easier to understand and work with.

There are many basis with respect to which we can represent a vector $x$ or a linear transformation $T$ and certain bases will allow us to represent the linear transformation $T$ in simpler forms. These simpler forms (such as diagonal matrices) are sometimes called canonical forms.

In computer graphics, changes of basis are frequently used to describe relative positions. For example, say we want to model a 3D person.

Now, we could certainly model this by figuring out the absolute translation, shear, and rotation for each part of the body. But that gets ugly very fast. We need to do lots of work to figure out where each piece of the body should go so that it all looks natural. And if we ever want to move any part of the body, we need to redo a lot of this calculation.

But what we usually want to talk about is the relative position. We don't want to talk about the position of the head in terms of the origin, we want to describe it in terms of the neck. To do so, we use a relative coordinate system, where the origin is at the neck. Then we just need to describe it with one position, shear, and rotation. And if the neck ever gets moved, the head (which is defined in terms of the neck) comes along for the ride.

So how do we get this relative coordinate system? With a change of basis! A change of basis is really just saying, "Pretend we're starting in the right place without any transformations applied. Then apply the transformations when we're done."

An important topic in LA is to express matrices in different basis and see what happens. Turns out that if you have a nice enough matrice you can express it as a diagonal matrice with respect to it's distinct eigenvectors. So you can decompose a matrice as $S^{-1}AS$ where $S$ is the change of basis and $A$ is a diagonal matrice. This has alot of uses, for example raising a matrice to an arbitrary power. Another application is SVD-decomposition, which is used in JPG compression.

You may be wondering this because you think of vector spaces as all being direct sums (probably finite) of copies of a field (probably $\Bbb R$ or $\Bbb C$.)

The problem is that you are not always presented with vector spaces that are generated by unit vectors. In particular, subspaces of vector spaces do not have to contain unit vectors of the containing space (say $\langle (1,2)\rangle\subseteq \Bbb R^2$). Now, that subspace certainly is isomorphic to a direct sum of copies of $\Bbb R$, but it would be awkward to relate to the original space if you insisted on calling things in the subspace by their $1$ dimensional coordinates.

Another example, already suggested in the comments, is that spaces of functions can be made into vector spaces, and yet they are not normally described as strings of coordinates.

From a more philosophical point of view, it's "better" to detach yourself from a particular frame of reference and just deal with the vector space and its vectors as abstract objects. Choosing a basis allows us to do a lot of practical calculations, but sometimes it is too "up close" to what we're doing, and we would be better off learning about vector spaces and transformations without worrying about what the strings of numbers are.

Being able to move between bases fluently frees us from the task of having to relate everything back to our own special basis, and prepares us to work in whatever frame of reference is convenient.