# on two dimensional graded vector spaces

Consider a graded vector space $V$ with basis $\{a, b\}$ such that $a \in V^2$ and $b \in V^5$.

Does this mean that $V=\bigoplus_{i\geq 0}V^i$ such that all $V^i$ are $0$ except $V^2$ and $V^5$ hence we can simply write $V=V^2\oplus V^5$ and if yes why don't we say directly that $V$ is a two dimensional vector space that can be written as the direct sum of two vector spaces $V=A\oplus B$ where $A$ is a one dimensioanl vector space with basis $\{a\}$ and $B$ is a one dimensional vector space with basis $\{b\}$.

Graded vector spaces need to retain their grading information. It might be that $V=V^1\oplus V^2$ is isomorphic as a vector space to $V'=V'^1\oplus V'^3$ but they are not isomorphic as graded vector spaces because an isomorphism would have to respect the degrees of elements in the grading. If were to just say $V$ is two-dimensional and isomorphic to the direct sum of $A$ and $B$, as you mention, then you have lost the graded structure.

We can write $V = V^2\oplus V^5$, but writing as in the second part of your question does not respect the grading. If we just write $V$ is two dimensional, or $V = A \oplus B$ where $A$ and $B$ are one-dimensional we have lost information about the grading and the homogeneous elements of $V$ (that is the multiples of $a$ and $b$).