## "Usefulness" of Basis for a Vector Space, General?

Hi, Everyone:

I am teaching an intro class in Linear Algebra. During the section on "Basis and Dimension"
a student asked me what was the use or purpose of a basis for a vector space V.

All I could think of is that bases allow us to define a linear map L for all vectors, once
we know the value of L at the basis vectors for V, i.e., vector spaces are free in their
bases and so on. I mumbled something about identifying all vector spaces over the
same field by their dimension, i.e., if V,V' v.spaces over F both, with the same dimension,
then they are isomorphic.

Are there other aspects where bases are equally important or more?

Thanks.
 Recognitions: Science Advisor Make sure students appreciate that "linear transformations" can include motions like rotations about the origin, reflections, shear, etc. They can even include translations if you use a specialized coordinate system ("projective coordinates"). Students are liable to confuse linear transformations with linear equations . The representation of linear transformations as matrices depends on using a basis - OK,maybe your students aren't sold on matrices either. A "dramatic" use of the use of a basis in a vector space is expressing a function as a sum of other functions. For example, fourier series, Chebeshev polynomials. Dont' forget , the wider meaning of "vector space" goes beyond operations on n-tuples of numbers. (See the thread: "a question on orthogonality relating to fourier analysis and also solutions of PDES" )

 Quote by Stephen Tashi A "dramatic" use of the use of a basis in a vector space is expressing a function as a sum of other functions. For example, fourier series, Chebeshev polynomials.
Careful, you need Schauder for that. On the topic of Fourier, each piano key corresponds to a sine function. The collection of finite sums gives the set of all piano chords.

There is also Lattice based crypto, but that is discrete so ymmv.

Recognitions:

## "Usefulness" of Basis for a Vector Space, General?

The most important thing for applications in physics, engineering, etc is when you can choose a basis where the matrices involved become diagonal.

But you won't have good examples of that until you have studied eigenvalues and eigenvectors.

Recognitions: