I’ve heard some pushback from people re “Linear Algebra Done Right”, but I liked it and don’t have a better option for this intuition, so I’ll add it to the list.
re: Axler’s textbook above, also check out the paper it’s based on which is just 18 pages, Down with determinants! (I know you know this, just for others’ edification). Abstract:
This paper shows how linear algebra can be done better without determinants. The standard proof that a square matrix of complex numbers has an eigenvalue uses determinants. The simpler and clearer proof presented here provides more insight and avoids determinants. Without using determinants, this allows us to define the multiplicity of an eigenvalue and to prove that the number of eigenvalues, counting multiplicities, equals the dimension of the underlying space. Without using determinants, we can define the characteristic and minimal polynomials and then prove that they behave as expected. This leads to an easy proof that every matrix is similar to a nice upper-triangular one. Turning to inner product spaces, and still without mentioning determinants, this paper gives a simple proof of the finite-dimensional spectral theorem.
Link: 3Blue1Brown: The determinant | Chapter 6, Essence of linear algebra
Also Linear Algebra Done Right by Sheldon Axler
Idea: The determinant of a matrix tells you the (signed) volume of a unit cube after applying the matrix transformation
Creator: Grant Sanderson (3Blue1Brown), Sheldon Axler
Reason: This geometric interpretation makes properties that seem arbitrary in formula-based definitions suddenly obvious. For example,
Why det(AB) = det(A) det(B)
Why det(A^T) = det(A)
Why det(A^-1) = 1/det(A)
Why det(kA) = k^n det(A)
Why det(A) = 0 means a matrix is not invertible
Why rank < n means det = 0
Why swapping matrix rows multiplies the determinant by −1
I’ve heard some pushback from people re “Linear Algebra Done Right”, but I liked it and don’t have a better option for this intuition, so I’ll add it to the list.
re: Axler’s textbook above, also check out the paper it’s based on which is just 18 pages, Down with determinants! (I know you know this, just for others’ edification). Abstract: