We are discussing abstract vector spaces in linear algebra class. So, I decided to do an application.
Let denote the polynomials of degree or less; the coefficients will be real numbers. Clearly is dimensional and constitutes a basis.
Now there are many reasons why we might want to find a degree polynomial that takes on certain values for certain values of . So, choose . So, let’s construct an alternate basis as follows:
This is a blizzard of subscripts but the idea is pretty simple. Note that and if .
But let’s look at a simple example: suppose we want to form a new basis for and we are interested in fixing values of .
. Then we note that
Now, we claim that the are linearly independent. This is why:
Suppose as a vector. We can now solve for the Substitute into the right hand side of the equation to get (note: for ). So are linearly independent vectors in and therefore constitute a basis.
Example: suppose we want to have a degree two polynomial where . We use our new basis to obtain:
. It is easy to check that