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>but I don't think they're not truly fundamental concepts that you need to learn to understand linear algebra I'd argue that without a deep understanding of ra

>but I don't think they're not truly fundamental concepts that you need to learn to understand linear algebra I'd argue that without a deep understanding of rank and nullity you do not understand linear algebra. Sure, you can lean to move symbols and compute simple things, but that is not much of an understanding. I'd guess you have not moved into deeper things - then you'll realize you're missing fundamental understanding of linear algebra needed to move on. It's like claiming one has a fundamental understanding of calculus by being able to solve high school level integrals, but really doesn't understand deep relations between the main ideas of calculus. Sure you can write things down, but there is a major difference between that level of knowledge and what I'd call a fundamental understanding of calculus. >span of the coefficients ???? Care to link to the thing you're misnaming? I have no idea what you're talking about, and I even googled the phrase. This is what I mean by using a proper book to learn from. >Aren't we discussing one? Nope - the book in question is not offered as a beginner course stats anywhere I am aware of. Care to show one? Calling it one then claiming it is evidence of one is simply circular logic. >Is that what the book does? So you have not looked at the book yet are arguing what kind of book it is? That about sums this up.

wizzwizz4 5y ago
> I have no idea what you're talking about, and I even googled the phrase. A linear transformation of some n-tuple (let's call it a) to some m-tuple (let's call it b) can be thought of as the sum of an element-wise multiplication of each item of a by an m-tuple coefficient (let's call the coefficient c_i). So b = ∑ a_i × c_i. The coefficients may or may not be linearly independent. The span (https://en.wikipedia.org/wiki/Linear_span https://en.wikipedia.org/wiki/Linear_span) of these coefficients is a useful property of the linear transformation; I consider it a more fundamental, more useful concept than your "rank". The concept of “rank” drops out of the (misnamed) “Gaussian Elimination” algorithm, and there are lots of theorems that involve it (probably because it was discovered early on in the development of this field), but it seems a rather complicated and unintuitive concept, to me. If you're using “rank”, you need to use a lot of theorems and lemmas and conversions between different representations of things that just don't seem necessary. I am happy to be corrected. > Nope - the book in question is not offered as a beginner course stats anywhere I am aware of. That doesn't mean it's not a textbook for beginners. It looks like one, to me. > So you have not looked at the book yet are arguing what kind of book it is? It's over 500 pages long. I've scrolled through it a bit, and nothing jumped out as obviously wrong; I haven't read it. (Though I did see a few different distributions named, hence my confusion.)
- ChrisLomont 5y ago
  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...

wizzwizz4 5y ago

> I have no idea what you're talking about, and I even googled the phrase. A linear transformation of some n-tuple (let's call it a) to some m-tuple (let's call it b) can be thought of as the sum of an element-wise multiplication of each item of a by an m-tuple coefficient (let's call the coefficient c_i). So b = ∑ a_i × c_i. The coefficients may or may not be linearly independent. The span (https://en.wikipedia.org/wiki/Linear_span https://en.wikipedia.org/wiki/Linear_span) of these coefficients is a useful property of the linear transformation; I consider it a more fundamental, more useful concept than your "rank". The concept of “rank” drops out of the (misnamed) “Gaussian Elimination” algorithm, and there are lots of theorems that involve it (probably because it was discovered early on in the development of this field), but it seems a rather complicated and unintuitive concept, to me. If you're using “rank”, you need to use a lot of theorems and lemmas and conversions between different representations of things that just don't seem necessary. I am happy to be corrected. > Nope - the book in question is not offered as a beginner course stats anywhere I am aware of. That doesn't mean it's not a textbook for beginners. It looks like one, to me. > So you have not looked at the book yet are arguing what kind of book it is? It's over 500 pages long. I've scrolled through it a bit, and nothing jumped out as obviously wrong; I haven't read it. (Though I did see a few different distributions named, hence my confusion.)

ChrisLomont 5y ago
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...