Can anyone explain in a simple approach, what basis polynomials are?
for example, what is Bernstein basis polynomials when we are saying that The $ n+1 $ Bernstein basis polynomials of degree $ n $, for $ n \in \mathbb{N}_{0} $, are defined as
$$ B_{k,n} (x) = {n \choose k} x^{k} (1 - x)^{n-k},\text{for } k \in \{0,1,...,n\} $$
It is appreciated if the answer comes with a visual approach.
In linear algebra a basis for a vector space is given by a set of vectors which decide where the axes of coordinates go in this space.
If we view a vector space of polynomials, what we do is we stuff the coefficients in the trivial basis (monomial basis for polynomial vector spaces) into vectors.
Basis polynomials are then the polynomials spanning the basis of the vector space of polynomials. Like vectors deciding where the coordinate axes go. For example if we expand $(1-x)^2,2x(1-x),x^2$ and stuff coefficients into vectors:
$$\left\{\begin{bmatrix}1\\-2\\1\end{bmatrix},\begin{bmatrix}0\\2\\-2\end{bmatrix},\begin{bmatrix}0\\0\\1\end{bmatrix}\right\}$$ These are the basis vectors of the Bernstein space of polynomials up to order 2, and the corresponding polynomials we calculated coefficients from are the basis polynomials which are scalar products like this:
$$(1-x)^2=[1,x,x^2]\cdot \begin{bmatrix}1\\-2\\1\end{bmatrix},2x(1-x)=[1,x,x^2]\cdot\begin{bmatrix}0\\2\\-2\end{bmatrix},x^2=[1,x,x^2]\cdot\begin{bmatrix}0\\0\\1\end{bmatrix}$$