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I have to show that the space of symmetric, the antisymmetric and the alternating bilinear forms each form a vector subspace of the space of all bilinear forms $\operatorname{Bil}(V,K)$ with $V$ being an $n$-dimensional $K$-vector space. Since the three spaces are all nonempty and subspaces of $\operatorname{Bil}(V,K)$, I think that all I have to show is that each space is closed under vector-addition and scalar-multiplication.

My idea was, since every bilinear form can be written as a matrix, to show, that for example symmetric matrices are closed under addition and multiplication, but they aren't under multiplication, so now I don't really know where the problem is with looking at the matrices or how to tackle the problem some other way.

Any ideas? Thanks in advance!

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Recall that on a vector space, there is generally no notion of multiplication of two vectors, only addition and multiplication by scalar. That is, given two symmetric bilinear forms $g_1,g_2 \colon V \times V \rightarrow K$ (or two symmetric matrices $A_1, A_2 \in M_n(K)$ representing bilinear forms), you only need to show that $g_1 + g_2$ and $cg_1$ are symmetric bilinear forms ($A_1 + A_2$ and $cA_1$ are symmetric matrices), where $c \in K$.

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