I'm confused about the essence of Gaussian elimination. Suppose there is a linear map between V and W. $T: V\rightarrow W$. Then there can be a matrix associated with the linear map $T$. Then apply Gaussian elimination to that matrix.
Probably the result will be a new matrix, but what does this new matrix mean?
Also, does Gaussian elimination have to do with the change of basis?
(If it does, then when using Gaussian elimination to solve linear equations, how to interpret this process of solving linear equations in terms of changing basis? Also, Gaussian elimination can also be used to determine whether a list of vectors is linearly dependent, how to interpret this in terms of changing the basis?)