Reply to a question below on why we care about the 4 fundamental subspaces in the study of Ax=b: In professor Strang words (not in this lecture) "understanding the four fundamental subspaces elevates the understanding of Ax=b to a vector space point of view, specifically, to to a vector subspace level of understanding"It will pay off when learning different types of matrix factorizations, for ex, Single Value decomposition, SVD, where A=U∑V^T.  Where U is composed of orthonormal vectors in the row space, V of orthonormal vectors in the Column Space,  ∑ by the diagonal matrix of the squares of the singular values.Similarly,  all other matrix factorizations QR, Q(LAMBDA)Q^T, etc benefit from their description in terms of subspaces.This understanding/insight is super powerful, even critical, in Optimization,Machine Learning,  ill posed problems, etc Plz don't disregard this Vector Subspace approach, it's only apparently that it doesn't seem to shine when being learnt for the first time, but believe me, it's pure gold, and the clarity with which  Professor Strang explains it is invaluable...he said it himself, it bends your mind when extended to sets of matrices, and to sets of functions (the differential equation example, a prelude to Fast Fourier Transform) but the dividends you gain are huge####[['fundamental subspaces', 'Mathematical_Related_Concept', 'POS']]