i would like to understand geometrical interpretation of Cokernel,from Wikipedia it is defined by the following way

cokernel (plural cokernels)
(category theory) The dual object of a kernel

i dont understand if it is related to linear algebra as well,because in linear algebra kernel is set of vector $v$ for given matrix $A$, for which


also we know that dual space ,for example dual vector space takes input as a vector and produce scalars,so according to these two definition,what does Cokernel in vector space?if it takes vector and produce scalar,then how above equation will change?thanks in advance


1 Answer 1


In general, the cokernal of a homomorphism $f:X\to Y$ of algebraic objects is defined to be $Y$ mod the image of $f$. In the case of a linear transformation given by a matrix $A:R^n\to R^m$, you can use the dot product in $R^m$ to identify $R^m/{\rm image\,}(A)$ with the orthogonal complement of the image. But this becomes more down to earth as follows: a vector $y$ is in $({\rm image\,}(A))^\perp$ when $y\cdot Ax=0$ for every vector $x$ in $R^n$. But $y\cdot Ax = (A^Ty)\cdot x$, and this is zero for all $x$ only when $A^Ty=0$. Equivalently $y^TA=0$.

Short version: kernel of $A$ is all $x$ with $Ax=0$, and cokernel of $A$ is all $y$ with $y^TA=0$.


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