Recap · Reading
What you can now do.
For any matrix, say which equations can be solved, how many solutions they have, and what they look like.
dim C(Aᵀ) + dim N(A) = n dim C(A) + dim N(Aᵀ) = m
- §3.1 Subspaces. Rails through the station: sets you cannot leave by adding and scaling.
- §3.2 The nullspace. Engines running, boat standing still: every x with Ax = 0.
- §3.2 Special solutions. The mixing desk: free variables are yours, pivot variables follow. n − r special solutions span N(A).
- §3.3 The complete solution. A route plus a loop: x = x_p + x_n.
- §3.4 Basis and dimension. A minimal toolkit. Every basis has the same size.
- §3.5 The four subspaces. The shadow: row space and nullspace among inputs, column space and left nullspace among outputs, sizes r, n − r, r, m − r.
Next lesson: orthogonality and least squares (chapter 4)
Perpendicular subspaces, projections, and the best answer to Ax = b when there is no exact one. That is how you fit a line to noisy sensor data.
Read along
Gilbert Strang, Introduction to Linear Algebra, 5th edition (2016), chapter 3. Lectures 6 to 10 of MIT 18.06 on MIT OpenCourseWare. Lecture 10 is the four subspaces.