Lecture Notes For Linear Algebra Gilbert Strang

The number of vectors in a basis. The dimension of the Column Space and Row Space is always equal to the Rank ( ) of the matrix. 4. Unit 3: Orthogonality and Least Squares When systems have more equations than variables (

) are the most important matrices in applied mathematics. They possess remarkable properties summarized by the Spectral Theorem: Their eigenvalues are always real numbers. Their eigenvectors are always perpendicular (orthogonal). They can be diagonalized using an orthogonal matrix , meaning . The Singular Value Decomposition (SVD) lecture notes for linear algebra gilbert strang

Most textbooks start with the "how"—how to multiply matrices or how to find a determinant. Strang starts with the . The number of vectors in a basis

For square matrices, we look for special vectors that do not change direction when multiplied by . They only scale. The Eigenvector Equation Unit 3: Orthogonality and Least Squares When systems

Always visualize how vectors in the row space map to the column space.

reveals exactly how many independent pieces of information exist inside your system.

). It forms the backbone of modern data science algorithms, including Principal Component Analysis (PCA) and image compression.