Kunquan Lan -fourth Edition- Pearson 2020 | Linear Algebra By

The Google PageRank algorithm is a great example of how Linear Algebra is used in real-world applications. By representing the web as a graph and using Linear Algebra techniques, such as eigenvalues and eigenvectors, we can compute the importance of each web page and rank them accordingly.

This is where weak students often stumble. Lan’s approach is methodical: he spends 15 pages on the definition of a vector space using concrete examples (R^n, polynomial spaces, matrix spaces) before introducing subspaces. The includes a set of “quick checks”—short true/false questions embedded in the margin—that force active reading. For instance: “Is the set of all 2x2 invertible matrices a subspace of M_22?” (Answer: No, it fails closure under addition.) Linear Algebra By Kunquan Lan -fourth Edition- Pearson 2020

The crown jewel of any linear algebra course. Lan uses a dynamical systems hook: “If you apply a matrix repeatedly to a vector, which direction dominates?” From there, he builds the characteristic polynomial, diagonalization, and the Cayley-Hamilton theorem. The Fourth Edition adds a subsection on (Google’s PageRank as a motivating example) and complex eigenvalues in rotation-scaling matrices. The Google PageRank algorithm is a great example

Designed for advanced courses, this chapter extends orthogonality to function spaces, introducing Fourier series mot`ivation. Lan wisely marks this as optional; most one-semester courses stop at Chapter 6. Lan’s approach is methodical: he spends 15 pages

$v_0 = \beginbmatrix 1/3 \ 1/3 \ 1/3 \endbmatrix$

Step-by-step proofs paired with numerical examples.