Given $x,y \in \mathbb{F}_2^n$. We define $x\times y = \sum_{i}x_iy_i$. Prove that the subspace $\{h\in \mathbb{F}_2^n: h\times x = 0\}$ has dimension $n-1$. I am trying for example to mount a basis for that subspace, supossing that it has dimension $n$ but I do not have any success. Couldyo help me please?
Special Vector space of dimension $n-1$
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linear-algebra
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1Instead of doing this, think about the map (fixing $x \in \mathbb F_2^n$) $\phi : h \to h \times x$, which goes from $\mathbb F_2^n$ to $\mathbb F_2$. Now, use rank-nullity theorem. – 2017-02-14
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0@астонвіллаолофмэллбэрг why use a map? this a space a not a map – 2017-02-14
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0Yes, but the rank-nullity theorem helps you understand the dimensions of the spaces with respect to the nature of the map you are using. Write down the rank nullity theorem, and you will see why. – 2017-02-14