Linear Algebra
GATE MA's Linear Algebra section tests vector spaces over R and C, linear transformations, matrix canonical forms, and inner-product geometry. Expect numerical rank and eigenvalue problems plus short proofs.
Core rules
- Rank-nullity: for T: V→W, dim V = nullity(T) + rank(T).
- Eigenvalues solve det(A−λI)=0; geometric multiplicity ≤ algebraic multiplicity.
- Cayley–Hamilton: p_A(A)=0; minimal polynomial divides p_A.
- A is diagonalizable iff the minimal polynomial has distinct linear factors; otherwise use Jordan form.
- For Hermitian A, eigenvalues are real and eigenvectors orthogonal; Gram–Schmidt orthonormalizes.
- Subspace tests require closure under addition and scalar multiplication; span and linear independence are frequent.
- The determinant is multilinear and alternating; row operations change it only by a known factor.
Worked example
Let A = [[4,1],[2,3]]. Characteristic polynomial: (4−λ)(3−λ)−2 = λ²−7λ+10 = (λ−5)(λ−2). For λ=5, (A−5I)v=0 gives [[−1,1],[2,−2]]v=0, so v=(1,1). For λ=2, [[2,1],[2,1]]v=0 gives v=(1,−2). Thus A=P diag(5,2) P^{-1} with P columns these eigenvectors; check P^{-1}AP.
Traps
- det(A+B) ≠ det A + det B; rank(AB) ≤ min(rank A, rank B).
- Algebraic multiplicity > geometric means not diagonalizable over R, even if all eigenvalues are real.
- For complex inner products, ⟨u,v⟩ is conjugate-linear in the second slot; orthogonal matrices become unitary.
- Do not confuse rank with the number of nonzero rows before row reduction.