I am recently learning from Loring W. Tu's *An Introduction to Manifolds* the concept graded algebra, which is used for introducing exterior algebra. I don't understand the following definition:

An algebra $A$ over a field $K$ is said to be graded if it can be written as a direct sum $A=\bigoplus_{k=0}^{\infty}A^k$ of vector spaces over $K$ such that the multiplication map sends $A^k\times A^l$ to $A^{k+l}$.

Here are the questions:

- What does $k$ in $A^k$ mean? Is it a superscript or the power of $A$? (If it is the power, what does $A^0$ mean?)
- Since I don't know much but some very basic knowledge in abstract algebra, I am trying to understand the concept with some simple examples. (I don't quite understand the wiki article of graded algebra). Is there any example as simple as the linear space ${\mathbb R}^n$ for the graded algebra? And what does it look like concretely?