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Matrices are linear maps, and matrix multiplication composes the linear maps. Now statements like the determinant of the product is the product of the determina

by tvural 10y ago

Matrices are linear maps, and matrix multiplication composes the linear maps. Now statements like the determinant of the product is the product of the determinants, the trace is preserved under similarity transforms (since similarity transforms just rewrite the linear map in a different basis), etc. are intuitive. I've always felt that these explicit calculations don't really get to the point. You can memorize them and still not really understand what's going on.

HQvIdsGkVw 10y ago
When I was a math student, we had two semesters of linear algebra. We only saw matrices at the end of the first semester; other than that it was all about linear mappings. The thing is, most linear algebra classes that are taught to other sections (biology, CS, …) start directly with matrices and are mostly computational.
gunnihinn 10y ago
Maybe you know something I don't, but I don't find these statements about the determinant and trace intuitive just from knowing that matrix multiplication corresponds to composition of linear maps. They can be made intuitive by appealing to geometry, but I don't see how to do that just by staring at the algebraic picture. Actually, proving them without picking a basis and then showing that what you get is invariant under base change is quite nontrivial and involves throwing a lot of heavy machinery around; see for example Coffman's non-coordinate proof of the statement about the trace in http://users.ipfw.edu/CoffmanA/pdf/book.pdf http://users.ipfw.edu/CoffmanA/pdf/book.pdf For the determinant, if you want to do everything without picking a basis, you're basically proving that the exterior power operation defines a functor on the category of finite-dimensional vector spaces, which isn't that bad to do, but you somehow have to explain what all those words mean along the way.

by tvural 10y ago

HQvIdsGkVw 10y ago

When I was a math student, we had two semesters of linear algebra. We only saw matrices at the end of the first semester; other than that it was all about linear mappings. The thing is, most linear algebra classes that are taught to other sections (biology, CS, …) start directly with matrices and are mostly computational.

gunnihinn 10y ago

Maybe you know something I don't, but I don't find these statements about the determinant and trace intuitive just from knowing that matrix multiplication corresponds to composition of linear maps. They can be made intuitive by appealing to geometry, but I don't see how to do that just by staring at the algebraic picture. Actually, proving them without picking a basis and then showing that what you get is invariant under base change is quite nontrivial and involves throwing a lot of heavy machinery around; see for example Coffman's non-coordinate proof of the statement about the trace in http://users.ipfw.edu/CoffmanA/pdf/book.pdf http://users.ipfw.edu/CoffmanA/pdf/book.pdf For the determinant, if you want to do everything without picking a basis, you're basically proving that the exterior power operation defines a functor on the category of finite-dimensional vector spaces, which isn't that bad to do, but you somehow have to explain what all those words mean along the way.