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?)

  • $\begingroup$ Your setup for using Gaussian elimination doesn't capture the usual circumstances for the "meaning" of it. While Gaussian elimination can be applied to any matrix, to produce a row-echelon matrix or (uniquely) a reduced row-echelon form, the meaning of such an exercise depends on the significance of the matrix. Are you familiar with representing a system of linear equations with an augmented matrix? $\endgroup$
    – hardmath
    Dec 7 '21 at 16:09

The matrix, say, $A$, depends not only on $T$ but also on the choice of bases in $V, W$. The $i$-th column of $A$ consists of coefficients of the linear combination of the basis of $W$ which is equal to the image of the $i$-th vector of the basis of $V$.

Row transformations of $A$ correspond to changes of the basis of $W$. Say, switching two rows, corresponds to switching two elements of the basis. So the Gauss reduction gives a better basis of $W$. The new matrix corresponds to the same $T$ and the two new bases (the old basis of $V$ and the new basis of $W$).

Similarly, column transformations of $A$ correspond to changes of the basis of $V$.

  • $\begingroup$ Then actually the Gaussian elimination is a way of changing the basis of codomain $W$, but we just don't care what this "new" basis will be, we are only interested in the RRE matrix. After Gaussian elimination, it's possible to then apply column operations to get a matrix only contains ones and zeros. At this time, the basis of domain and codomain are both changed even though we don't know what exactly the new basis for domain and codomain are, but at this time the number of ones is the rank of the matrix. Is this correct? $\endgroup$
    – Yink
    Nov 27 '21 at 5:07
  • $\begingroup$ That is correct. Only the rank can be determined even before column transformations: the number of non-zero rows in RREF. $\endgroup$
    – markvs
    Nov 27 '21 at 5:18

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