4 ms·

In my experience, it's a little bit easier for new students to understand that the image of a matrix is the span of its columns, hence column space.

by noqc 3y ago

CamperBob2 3y ago
Perhaps, but that's about as useful as pointing out that monads are a monoid in the category of endofunctors. What's the "image of a matrix?" Coming at LA from a 3D graphics background, I've never heard that term before. And what does the "span of its columns" mean? To me, each column represents a different dimension of the basis vector space, so the notion that X, Y, and Z might form independent "column spaces" of their own is unintuitive at best. These are all questions that can be Googled, of course, but in the context of a coherent, progressive pedagogical approach, they shouldn't need to be asked. And they certainly don't belong in the first chapter of any introductory linear algebra text, much less the preface.
- Koshkin 3y ago
  In the context of linear algebra, a matrix is a linear map. A map is characterized by its domain and its image. These are very important characteristics.
- ravi-delia 3y ago
  > To me, each column represents a different dimension of the basis vector space, so the notion that X, Y, and Z might form independent "column spaces" of their own is unintuitive at best. I can't help but feel a treatment of linear algebra that assumes all matrices are invertible by default isn't a very good treatment at all. Column spaces are exactly how you harness your (very useful!) intuition that the columns of a matrix are where the basis goes. I agree that it should be defined before use, but it is- in the textbook proper. The preface is for the author to express themselves! Now row spaces are an abomination, but that's because I'm not really a computation guy. I'm sure they're great if you get to know them.

4 ms·

In my experience, it's a little bit easier for new students to understand that the image of a matrix is the span of its columns, hence column space.

by noqc 3y ago

In my experience, it's a little bit easier for new students to understand that the image of a matrix is the span of its columns, hence column space.

CamperBob2 3y ago
Perhaps, but that's about as useful as pointing out that monads are a monoid in the category of endofunctors. What's the "image of a matrix?" Coming at LA from a 3D graphics background, I've never heard that term before. And what does the "span of its columns" mean? To me, each column represents a different dimension of the basis vector space, so the notion that X, Y, and Z might form independent "column spaces" of their own is unintuitive at best. These are all questions that can be Googled, of course, but in the context of a coherent, progressive pedagogical approach, they shouldn't need to be asked. And they certainly don't belong in the first chapter of any introductory linear algebra text, much less the preface.
- Koshkin 3y ago
  In the context of linear algebra, a matrix is a linear map. A map is characterized by its domain and its image. These are very important characteristics.
- ravi-delia 3y ago
  > To me, each column represents a different dimension of the basis vector space, so the notion that X, Y, and Z might form independent "column spaces" of their own is unintuitive at best. I can't help but feel a treatment of linear algebra that assumes all matrices are invertible by default isn't a very good treatment at all. Column spaces are exactly how you harness your (very useful!) intuition that the columns of a matrix are where the basis goes. I agree that it should be defined before use, but it is- in the textbook proper. The preface is for the author to express themselves! Now row spaces are an abomination, but that's because I'm not really a computation guy. I'm sure they're great if you get to know them.