Linear Algebra Practice Questions

### 1. Definition Study of vectors, vector spaces, linear transformations, and systems of linear equations represented through matrices. ### 2. Matrix Operations & Properties - (AB)ᵀ = BᵀAᵀ - (AB)⁻¹ = B⁻¹A⁻¹ - det(AB) = det(A)·det(B) - det(Aᵀ) = det(A) - det(A⁻¹) = 1/det(A) - det(cA) = cⁿ·det(A) for n×n matrix ### 3. Rank (Very Important) - Rank = number of non-zero rows in row echelon form = number of linearly independent rows (= columns) - rank(A) + nullity(A) = n (number of columns) — Rank-Nullity theorem - rank(AB) ≤ min(rank(A), rank(B)) - rank(A + B) ≤ rank(A) + rank(B) ### 4. System of Linear Equations Ax = b (Very Important) | Condition | Solution | |---|---| | rank(A) = rank([A|b]) = n | Unique solution | | rank(A) = rank([A|b]) < n | Infinitely many solutions | | rank(A) ≠ rank([A|b]) | No solution (inconsistent) | For homogeneous system Ax = 0: - Always has trivial solution x = 0 - Non-trivial solution exists iff rank(A) < n iff det(A) = 0 ### 5. Eigenvalues & Eigenvectors (Very Important) Ax = λx → (A − λI)x = 0 → det(A − λI) = 0 (characteristic equation) **Properties of Eigenvalues**: - Sum of eigenvalues = trace(A) = Σaᵢᵢ - Product of eigenvalues = det(A) - Eigenvalues of Aᵏ = λᵏ; of A⁻¹ = 1/λ; of Aᵀ = same as A - If A is triangular → eigenvalues are diagonal entries - Symmetric matrix → all eigenvalues real - Skew-symmetric → eigenvalues purely imaginary or zero ### 6. Cayley-Hamilton Theorem Every matrix satisfies its own characteristic equation. If p(λ) = 0 is characteristic equation → p(A) = 0. Use: find A⁻¹ or higher powers of A without direct computation. ### 7. Diagonalization A is diagonalizable if it has n linearly independent eigenvectors. A = PDP⁻¹ where D = diagonal matrix of eigenvalues, P = matrix of eigenvectors. Symmetric matrices are always diagonalizable. ### 8. Vector Spaces - **Subspace**: Closed under addition and scalar multiplication, contains zero vector - **Basis**: Linearly independent set that spans the space - **Dimension**: Number of vectors in basis - **Column space**: Span of columns of A; dimension = rank(A) - **Null space**: All x such that Ax = 0; dimension = nullity(A) ### 9. Special Matrices | Matrix | Property | |---|---| | Symmetric | A = Aᵀ | | Skew-symmetric | A = −Aᵀ, diagonal entries = 0 | | Orthogonal | AᵀA = I, so A⁻¹ = Aᵀ, det = ±1 | | Idempotent | A² = A, eigenvalues are 0 or 1 | | Nilpotent | Aᵏ = 0 for some k, all eigenvalues = 0 | ### 10. GATE Trick Trace = sum of eigenvalues and det = product of eigenvalues eliminates need to solve full characteristic equation when two eigenvalues are known. For 2×2 matrix: characteristic equation is λ² − (trace)λ + det = 0. Rank-nullity theorem is used in almost every system-of-equations question.