I'm following this intro to geometric algebra and I'm a bit confused about what the difference between these two objects is.
The author states (p. 12)
Let r > 1; then an r-blade or simple r-vector is a product of r orthogonal (thus anticommuting) vectors. A finite sum of r-blades is called an r-vector or homogeneous multivector of grade r.
So by this I can infer that an r-blade is the same as a simple r-vector, but not necessarily an r-vector.
But my problem is then, what is meant by finite sum? If I have a sum of a single r-blade, then that seems to meet the criterion for being an r-vector. So it seems like every r-blade is an r-vector.
Based on my understanding of GA, $r$-vectors are linear combinations of basis $r$-blades. Basis $r$-blades are products of $r$ linearly independent basis vectors that span an $r$-dimensional subspace embedded in some $d$-dimensional space (where $r \leq d$). For such a $d$-dimensional space, one can construct $\left( \begin{array}{c}d\\r\end{array}\right)$ ($d$ choose $r$) independent $r$-blades. Based on the choice of the basis vectors the constructed $r$-blades will span different oriented sub-spaces. If a specific $r$-vector is not fully embedded in one of these oriented sub-spaces, then it will have to be expressed as a linear combination of more than one $r$-blade. With that being said, another choice of $r$ basis vectors might allow for a single basis $r$-blade to span the specific sub-space containing the desired $r$-vector. This is similar to how one can redefine the set of basis vectors in a 3D space in such a way as to align a single one of those basis vectors (1-blade) with any arbitrary vector (1-vector) direction.