| source: knowledge_base | |
| topic: linear_algebra | |
| class_level: class_12 | |
| difficulty: advanced | |
| Linear Algebra Essentials: | |
| Matrix multiplication: row-by-column. Determinant 2x2: ad-bc. | |
| Inverse: A^(-1)=adj(A)/det(A), exists iff det(A)≠0. | |
| Eigenvalues: det(A-lambda*I)=0. Eigenvectors: solve (A-lambda*I)v=0. | |
| Rank-Nullity: rank(A)+nullity(A)=n. Dot product: u·v=|u||v|cos(theta). Orthogonal: u·v=0. |