# Linear algebra: Alternated multilinear forms

Let $A^{\#}:A_{n}(F)\rightarrow A_{n}(E)$ define by $A^{\#}f(v_{1},\cdots ,v_{n})=f(Av_{1},\cdots, Av_{n})$, where $A_{n}(F)$ is the space of the alternated n multilinear forms in F. Verify that $(\alpha A)^{\#}=\alpha (A^{\#})$, where $\alpha$ is a scalar and $A:E\rightarrow F$ is a linear transformation.

$(\alpha A)^{\#}f(v_{1},\cdots ,v_{n})=f(\alpha Av_{1},\cdots,\alpha Av_{n})=\alpha^{n}f(Av_{1},\cdots, Av_{n})\neq \alpha (A^{\#})f(v_{1},\cdots ,v_{n})$

then I showed that it's false, am I missinterpreting all?

a hint would be appreciated, thanks in advance .

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What is $\alpha$ supposed to be? A scalar? And $A$, is it supposed to be a linear transformation $A: E\to F$? – Willie Wong♦ Jun 30 '11 at 14:27
yes $\alpha$ is a scalar. Edited it – Ivan3.14 Jun 30 '11 at 14:28
yes I forgot,  A is linear transformation from E to F – Ivan3.14 Jun 30 '11 at 14:39
Is that the exact wording of the question? (and not something like, verify that pullbacks are linear, etc.)? – Dactyl Jun 30 '11 at 15:32
Unless the notation $\alpha(A^\sharp)$ means something special, instead of just scalar multiplication, I think what you did is correct. – Willie Wong♦ Jun 30 '11 at 18:46
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