# Exterior powers of vector space and kernel

Let $E$ be a vector space and $A$ a subspace of $E$. Let $q$ be a positive integer. Then we can define a subspace $\Lambda^q A$ of the $q$-th exterior power of $E$ by

$$\Lambda^qA=span\{\ a_1\wedge\dots\wedge a_q\ |\ a_1,\dots,a_q\in A\ \}$$

On the other hand, assume that $\varphi$ is a linear application from $E$ into $F$. We "exteriorize" $\varphi$ to obtain a linear application $\Lambda^q$ from $\Lambda^qE$ into $\Lambda^qF$ satisfying

$$\Lambda^q\varphi(e_1\wedge\dots\wedge e_q)=\varphi(e_1)\wedge\dots\wedge\varphi(e_q)$$

for all $e_1,\dots,e_q$ in $E$.

Now, if $A=Ker(\varphi)$, then $\Lambda^qA$ is a subspace of $Ker(\Lambda^q\varphi)$ but the inclusion is in general strict, since the decomposable elements of the form $a\wedge e_2\wedge\dots\wedge e_q$ are also in $Ker(\Lambda^q\varphi)$

My questions:

1. How can I describe $Ker(\Lambda^q\varphi)$ using $A$?
2. Is there a simple way to define a linear application $\Phi$, related to $\varphi$, and whose kernel is $\Lambda^qA$?

That's two questions but there are closely related, so I think it is worth having them at the same place.

The kernel of $\wedge^n \phi$ is precisely the antisymmetric subspace generated by the elements you describe. In particular, we have $$\ker \wedge^n\phi = A \wedge \left(\wedge^{n-1} E\right)$$
Perhaps you should consider the map $$\Phi = \phi \wedge \left(\wedge^{n-1} I\right)$$ where $I$ denotes the identity mapping over $E$.
This map is sometimes denoted as $A^{[n]}$. It is notable that $A^{[n]} = \frac {d}{dt} \wedge^{n}(I + tA)$. Bhatia discusses the map briefly in section I.5 of his "Matrix Analysis", but he doesn't say much else about it.