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


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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