# express skew-commutative product of Whitney sum of vector bundles in tensor products

Let $\xi$ and $\eta$ be vector bundles over a paracompact space $B$ and $\xi\oplus\eta$ be their Whitney sum. Can we write $\Lambda(\xi\oplus\eta)\cong \Lambda(\xi)\otimes \Lambda(\eta)$ as (graded) vector bundles?

For vector spaces $V,W$, we have $\Lambda(V\oplus W)\cong \Lambda(V)\otimes \Lambda(W)$ are isomorphic as graded vector spaces. But I am confused for vector bundle case...

• You just transplant what you know about vector spaces to the trivializations of your bundles, so you're done. :D – Ted Shifrin Apr 29 '14 at 3:56
• In Milnor's book (whose notation you seem to be using), on chap. 3, p. 31, he answers your question: how to transfer linear algebra constructions to vector bundles. It is a bit abstract, but very clear and complete, as is usual with Milnor. – Gil Bor May 1 '14 at 4:01