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 Aug 18 comment Are $\operatorname{Ker}(T)$ and $W/\operatorname{Im}(T)$ isomorphic? Thanks! What do "canonical" and"non-canonical"mean in this case and in its most generaility (I have seen them many times)? Aug 18 asked Are $\operatorname{Ker}(T)$ and $W/\operatorname{Im}(T)$ isomorphic? Aug 18 comment Transpose of a linear mapping hanks! (1) I was wondering in your reply to part 3, when talking about existence and uniqueness of adjoint, can the inner products on domain and codomain be relaxed to more general bilinear forms? (2) In the rephrase of the Fundamental theorem by you, is the second formula $\mathrm{Im}(T^*)=(\mathrm{ker}(T))^{\perp}$ supposed to be $\mathrm{Ker}(T^*)=(\mathrm{Im}(T))^{\perp}$ instead? Aug 18 accepted Transpose of a linear mapping Aug 18 asked orthogonal polarity and bilinear form Aug 18 comment Transpose of a linear mapping Thanks! Are the following correct? (1) Any two vector spaces with the same dimension are isomorphic. (2) The two definitions of transpose of a linear mapping don't depend on inner products, and they are pure vector space concepts. Aug 18 comment Transpose of a linear mapping I might be wrong before. But what I meant was the following. (1) Are there some isomorphic relation in the sense of vector spaces (without referring to inner product, and therefore not considering orthogonal complement) between images and kernels of $T$ and $T^*$. (2) Is it right that the two definitions of transpose of a linear mapping don't depend on inner products, and they are pure vector space concepts? Aug 18 comment Transpose of a linear mapping (1) Isn't that the first definition of transpose of a linear mapping depends on two bilinear forms but not inner products on domain and codomain? (2) Consider each kind of transpose definition and assume there are no inner products. Can coimage of $T$ and image of $T^*$ be isomorphic? Similar question for cokernel of $T$ and kernel of $T^*$? Aug 18 comment Transpose of a linear mapping The transpose of a linear mapping is a pure vector space concept, and doesn't require an inner product. So is there some theorem(s) that relates Image and kernels of $T$ and $T*$ or $T^T$? Aug 18 awarded Yearling Aug 17 comment Transpose of a linear mapping Thanks! As to 3, is there a similar theorem for vector spaces instead of for inner product spaces? Aug 17 revised Transpose of a linear mapping added 5 characters in body Aug 17 asked Transpose of a linear mapping Aug 17 accepted Integral of differential form and integral of measure Aug 17 accepted Definitions of direct product and of direct sum Aug 16 revised Definitions of direct product and of direct sum added 4 characters in body Aug 16 comment Definitions of direct product and of direct sum Thanks! From your update about direct sum, I think you now understand my questions. As to direct product, is it the same concept as product in category theory, and as Cartesian product in set theory? Aug 16 comment Definitions of direct product and of direct sum @Dylan: When opening the Wikipedia articles, I was hoping ideally I could see at the very beginning the definitions of direct sum/product at the most general level. However, all I saw are some vague description of their relations to (co)product in Category theory (sometimes coincide, but not always), followed by specific direct sum/product for Abelian groups, for vector spaces, .... From the replies and comments, I now guess that there are perhaps no such unified definitions of direct sum/product. Aug 16 comment Definitions of direct product and of direct sum Thanks! I understood the definitions of (co)product prior to asking the questions. What I was curious about is there seem to be no definition for direct sum/product which can unify the specific direct sums/products for Abelian groups, for vector spaces, ..., and which, as Wikipedia says, are not the same as (co)product in category theory. Aug 16 revised Definitions of direct product and of direct sum edited tags