# Relationship Between Blades and Tensors

It states that blades are a generalization of vectors. Fair enough.

But then in this link: https://math.wikia.org/wiki/Tensor

It states that tensors are a generalization of tensors.

Now that has me confused. It doesn't seem like blades and tensors are the same thing, but how can that be if they both generalizations of the same thing? Unless there is more than one way to generalize the same thing? Though, if possible, that in itself seems strange also.

Are blades a geometric generalization of vectors? While tensors are a matrix (algebraic?) generalization of vectors? But even if that's the case, that seems weird too since it seems like the definitions would quickly converge so having two different names for the same thing seems pointless.

• A generalization just means less specific. A rock is a generalization of a planet. A satellite is a generalization of a planet. Rocks and satellites are usually not related (though they can be, e.g. a planet). – NicNic8 Dec 10 '19 at 17:16
• @NicNic8 Seems counterproductive in math though if you generalize the same thing twice in two different directions. – DKNguyen Dec 10 '19 at 18:47
• Different generalizations give you different types of mat. For example, one generalization of the real numbers is vector spaces, and that gives you Linear Algebra. Another generalization is complex numbers, and that gives you complex analysis. – NicNic8 Dec 10 '19 at 18:50

More precisely, a blade is the tensor given by the skew-symmetric product of a set of vectors of a vector space $$V$$:
$$v_1 \wedge \cdots \wedge v_k = \sum_{\sigma \in \mathfrak{S}_k} (-1)^\sigma v_{\sigma(1)} \otimes \cdots \otimes v_{\sigma(k)}$$ where the sum is over the full permutation group on $$k$$ elements and $$(-1)^\sigma$$ is the sign of the permutation $$\sigma$$.
In this context, blades can also be interpreted as elements of the (affine cone over the) Grassmannian of $$k$$-planes in its Plucker embedding in $$\Lambda^k V$$.