[catnaroek]

Exterior algebras themselves are defined in terms of standard linear algebra concepts (tensor algebras, ideals, quotients). I don't see how you can claim something is an alternative to linear algebra if you need to know linear algebra to define it, nevermind use it.

[Retra]

Left is an alternative to right. Right is the opposite of left. They are alternatives, yet defined in terms of each other. You can't know one without the other.

Doesn't seem contradictory to me...