You concisely illustrate my point. You misuse words so that others that have learned the material cannot understand what you're writing. Calling an m-tuple a c

You concisely illustrate my point. You misuse words so that others that have learned the material cannot understand what you're writing. Calling an m-tuple a coefficient and then calling the column span the "coefficient span" does not show understanding of the material. >I consider it a more fundamental, more useful concept than your "rank". The dimension of the span is the rank. It's exactly why rank is important. The nullity is the dimension of the kernel, and rank + nullity = n. Saying rank is not important is like saying dimension is not important. The dimension of a vector space is the first and most important invariant that describes the space. Rank is that dimension for the image of a linear transform. It's absolutely fundamental. It's why when describing some linear algebra thing, one usually starts with something like "Let V be a n-dimensional real vector space" or similar. We rarely (bordering on never) write "Let V be a vector space spanned by the following vectors", and we never write "the coefficient span of the transform". >The concept of “rank” drops out of the (misnamed) “Gaussian Elimination” algorithm No, it is a fundamental property of the linear transformation. Gaussian elimination is but one of many ways to compute it. And no matter what choices you use in your elimination, you will always get the same rank. This is super important - that Gaussian elimination gives knowledge of the linear transformation that is independent of choice of basis or of steps performed. Changing of either vector space changes the numbers in the matrix, and different Gaussian elimination steps may have differing intermediate steps, but the rank is the rank is the rank. It is also true that the row span = the column span = the rank, which is also not immediately obvious. These are theorems proven in basic linear algebra, and fundamental to claiming to understand basic linear algebra. That rank shows up in Gaussian elimination is not some artifact or unique thing to Gaussian elimination. Since rank shows up everywhere, when it also shows up in Gaussian elimination shows that Gaussian elimination is doing something fundamental - it is but one of many, many ways that rank pops up over and over in linear algebra. Rank (and nullity) are absolutely fundamental. And no one calls that m-tuple a coefficient. None of this is in the book above. Zero. A final nice point to illustrate, here [1] is the index to Strang's Introduction to Linear Algebra. Rank occurs more than any just about every other entry in the index. Here's Serge Lang's book [2]. After the word dimension, rank occurs the most in the index. Book after book shows that after the concept of dimension, rank is probably the most important concept in linear algebra. I could go on and on. You cannot claim to understand linear algebra without know how pervasive and useful rank is. [1] https://math.mit.edu/~gs/linearalgebra/linearalgebra5_Index.pdf https://math.mit.edu/~gs/linearalgebra/linearalgebra5_Index.... [2] http://www.math.nagoya-u.ac.jp/~richard/teaching/f2014/Lin_alg_Lang.pdf http://www.math.nagoya-u.ac.jp/~richard/teaching/f2014/Lin_a...