Algebra Gilbert Strang __exclusive__ — Lecture Notes For Linear

Gilbert Strang’s 18.06 Linear Algebra lectures at MIT are legendary because they shift the focus from tedious matrix calculations to the beautiful geometric intuition behind the math.

10. Linear Transformations & Change of Basis

• Matrices as linear maps between vector spaces.
• Change of basis formula: [T]_B = P^-1 [T]_E P.
• Similarity, invariant subspaces, canonical forms.

MIT OpenCourseWare (OCW): The official home of 18.06. You can find PDF summaries of every lecture, often handwritten or typed by his TAs. lecture notes for linear algebra gilbert strang

Ax = b (no solution)
       ↓
Minimize ||Ax - b||^2
       ↓
Derivative = 0 → A^T A x̂ = A^T b
       ↓
If columns independent → x̂ = (A^T A)^-1 A^T b
       ↓
Projection p = A x̂

Strang’s most famous contribution to teaching is the "Big Picture" diagram involving four subspaces associated with any Column Space All linear combinations of the columns (in All solutions to All linear combinations of the rows (in Left Nullspace All solutions to Fundamental Theorem of Linear Algebra Gilbert Strang’s 18

11. Advanced Topics (selected)

• Jordan form (brief): when matrix not diagonalizable.
• Functions of matrices, matrix exponentials for ODEs.
• Conditioning and numerical stability: condition number cond(A) = ||A||·||A^-1||.
• Iterative methods: power method for dominant eigenvalue.