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I would teach linear algebra with a book that barely mentions linear transformations. In finite dimensions, linear transformations and matrices are exactly the
by jknz 9y ago
I would teach linear algebra with a book that barely mentions linear transformations.
In finite dimensions, linear transformations and matrices are exactly the same object mathematical objects, with very different notations (matrix notation (boxes with numbers inside) vs the linear space/linear transformation notation). I would rather the students to learn deeper mathematics only in matrix notation, rather than to master less substantial mathematics with both notations.
Teaching both notations may reinforce the idea that matrices and linear transformations are different mathematical objects in finite dimension. Teaching the more abstract notation is mainly useful in infinite dimension (Hilbert spaces).
- btilly 9y agoThis is exactly backwards. Whether you're discussing the Jacobian of a function, or change of basis matrices, learning the matrix formula is a lot less useful than seeing how it falls out of the linear function definition. The formula is hard to memorize and gives no intuition for why anything is true. But from the linear function definition it is easy to reconstruct the formula. In fact this is so true that I would say that anyone who only knows the matrix definition does not actually understand linear algebra.
- catnaroek 9y agoAnd you would forever cripple your students. As potential users of linear algebra, they would be mentally mutilated beyond all hope of regeneration.
- jacobolus 9y agoYou have succumbed to the coordinate virus, http://geocalc.clas.asu.edu/pdf/MathViruses.pdf http://geocalc.clas.asu.edu/pdf/MathViruses.pdf
- wolfgke 9y ago> In finite dimensions, linear transformations and matrices are exactly the same object mathematical objects They are not: Matrices represent linear transformations with respect to a given basis. Linear transformations are completely independent from any chosen basis.