Linear Algebra Notes & Formulas

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.