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**Problem:**Let [itex]L[/itex] and [itex]M[/itex] be finite dimensional linear spaces over the field [itex]K[/itex] and let [itex]g: L\times M \rightarrow K[/itex] be a bilinear mapping. Let [itex]L_0[/itex] be the left kernel of [itex]g[/itex] and let [itex]M_0[/itex] be the right kernel of [itex]g[/itex].

a) Prove that [itex]dim L/L_0 = dim M/M_0[/itex].

b) Prove that [itex]g[/itex] induces the bilinear mapping [itex]g': L/L_0 \times M/M_0 \rightarrow K, g'(l+L_0, m+M_0) = g(l,m)[/itex], for which the left and right kernels are zero.

I am trying to prove b) before a) (I think this would make a) relatively trivial). However, I am still getting used to the notion of quotient spaces as sets of equivalence classes. I am unsure as to how to deal with the bilinear mapping of equivalence classes. All that I have seen so far deal with vector spaces. Perhaps someone can give me some insight as to how to show that this mapping is well defined? How can I visualize this better? My professor is of little help, so I appreciate any insights anyone can give me.

Thanks all!

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