For anyone who is already familiar with the Prof. Strang's lectures from previous years, the main new thing in this five-lecture mini-series is he tries to condense the material even further—maximum intuition and power-ideas, instead of the full-length in-class lecture format with derivations. This makes the material difficult to understand for beginners, but makes a great second source in addition to or after a regular LA class.
One of the interesting new ways of thinking in these lectures is the A = CR decomposition for any matrix A, where C is a matrix that contains a basis for the column space of A, while R contains the non-zero rows in RREF(A) — in other words a basis for the row space, see https://ocw.mit.edu/resources/res-18-010-a-2020-vision-of-li...
Thinking of A as CR might be a little intense as first-contact with linear algebra, but I think it contains the "essence" of what is going on, and could potentially set the stage for when these concepts are explained (normally much later in a linear algebra course). Also, I think the "A=CR picture" is a nice justification for where RREF(A) comes about... otherwise students always complain that the first few chapters on Gauss-Jordan elimination is "mind-numbing arithmetic" (which is kind of true...) but maybe if we present the algorithm as "finding the CR-decomposition which will help you understand dozens of other concepts in the remainder of the course" it would motivate more people to learn about RREFs and the G-J algo.
Since y'all are code-literate, here the SymPy function for finding the CR-decomposition of any matrix A:
def crd(A):
"""
Computes the CR decomposition of the matrix A.
"""
rrefA, licols = A.rref() # compute RREF(A)
C = A[:, licols] # linearly indep. cols of A
r = len(licols) # = rank(A)
R = rrefA[0:r, :] # non-zero rows in RREF(A)
return C, R
One of the interesting new ways of thinking in these lectures is the A = CR decomposition for any matrix A, where C is a matrix that contains a basis for the column space of A, while R contains the non-zero rows in RREF(A) — in other words a basis for the row space, see https://ocw.mit.edu/resources/res-18-010-a-2020-vision-of-li...
Example you can play with: https://live.sympy.org/?evaluate=C%20%3D%20Matrix(%5B%5B1%2C...
Thinking of A as CR might be a little intense as first-contact with linear algebra, but I think it contains the "essence" of what is going on, and could potentially set the stage for when these concepts are explained (normally much later in a linear algebra course). Also, I think the "A=CR picture" is a nice justification for where RREF(A) comes about... otherwise students always complain that the first few chapters on Gauss-Jordan elimination is "mind-numbing arithmetic" (which is kind of true...) but maybe if we present the algorithm as "finding the CR-decomposition which will help you understand dozens of other concepts in the remainder of the course" it would motivate more people to learn about RREFs and the G-J algo.