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 .

So

. 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

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