Linear Algebra and Vector Geometry
Linear algebra concerns matrix structure, consistency of systems, independence, rank, eigenvalues and quadratic forms. The shared material below is mathematical foundation practice; spatial line and plane geometry is not presented as Mathematics III examination content.
Systems and rank
- Reduce the augmented matrix to row echelon form; count pivots in the coefficient block (rank A) and in the augmented block (rank [A|b]).
- Consistent iff rank A = rank [A|b]. Unique solution iff that common rank equals the number of unknowns; otherwise there are n − rank free parameters.
- Homogeneous systems always have the trivial solution. For a square coefficient matrix, nontrivial solutions exist exactly when det A = 0; in general use rank A less than the number of columns.
Determinants and eigenvalues
- det(AB) = det A · det B; A is invertible iff det A ≠ 0; a triangular matrix has determinant equal to the product of its diagonal entries.
- Eigenvalues solve det(A − λI) = 0; eigenvectors solve (A − λI)v = 0.
- Trace equals the sum of eigenvalues and determinant their product - a fast arithmetic check.
Independence and quadratic forms
A matrix with independent columns has a trivial nullspace. For a real symmetric matrix, orthogonal eigenvectors can form an orthonormal eigenbasis. A quadratic form is positive definite when its value is positive for every nonzero vector; eigenvalues provide a useful test for symmetric matrices. Always identify matrix dimensions before multiplying or applying a determinant.
Worked example
A = [[2, 1], [1, 2]]. det(A − λI) = (2 − λ)² − 1 = λ² − 4λ + 3 = (λ − 1)(λ − 3). Eigenvalues 3 and 1; check: trace 4, determinant 3. For λ = 3, (A − 3I)v = [[−1, 1], [1, −1]]v = 0 gives v = (1, 1). For λ = 1, [[1, 1], [1, 1]]v = 0 gives v = (1, −1). Because A is symmetric, the eigenvectors are orthogonal - confirm by dot product.
Traps that cost marks
- Reporting infinitely many solutions when rank [A|b] exceeds rank A; that system is inconsistent.
- Using det A = 0 to conclude 'no solution' - it means no unique solution.
- Normalising eigenvectors inconsistently; if a unit eigenvector is required, divide by its length and say so.
- Cross-product order: b×a = −a×b, so swapping silently flips the normal direction.
- Forgetting that a plane's normal vector can be read directly from its coefficients.