Linear Algebra Exam Quick Reference | Generated by AI
Quick reference for key concepts and formulas. Focus on matrices, determinants, linear systems, vector spaces, transformations, eigenvalues, and quadratic forms—core topics from past exams. Review definitions, properties, and computations.
1. Matrices
- Definition: \( A = [a_{ij}] \in \mathbb{R}^{m \times n} \).
- Operations:
- Addition: \( (A + B){ij} = a{ij} + b_{ij} \).
- Scalar: \( (cA){ij} = c a{ij} \).
- Multiplication: \( (AB){ij} = \sum_k a{ik} b_{kj} \) (if compatible sizes).
- Transpose: \( (A^T){ij} = a{ji} \); \( (AB)^T = B^T A^T \), \( (A^T)^T = A \).
- Inverse (square): \( AA^{-1} = I \); \( (AB)^{-1} = B^{-1} A^{-1} \); \( (A^T)^{-1} = (A^{-1})^T \).
- Types:
- Diagonal: Non-zero only on diagonal.
- Upper/Lower Triangular: Zeros below/above diagonal.
- Symmetric: \( A = A^T \).
- Orthogonal: \( A^T A = I \) (columns orthonormal).
2. Determinants (det A)
- Properties:
- \( \det(AB) = \det A \cdot \det B \); \( \det(A^T) = \det A \); \( \det(cA) = c^n \det A \).
- Row/Column swap: Multiplies by -1.
- Add multiple of row/column: No change.
- Scale row/column by c: Multiplies by c.
- \( \det I = 1 \); \( \det A = 0 \) if singular (dependent rows/columns).
- Computation:
- 2x2: \( \det \begin{pmatrix} a & b \ c & d \end{pmatrix} = ad - bc \).
- Cofactor Expansion (row i): \( \det A = \sum_j a_{ij} C_{ij} \), where \( C_{ij} = (-1)^{i+j} M_{ij} \) (minor det).
- Triangular: Product of diagonal entries.
- Adjugate/Inverse: \( A^{-1} = \frac{1}{\det A} \adj A \), where \( \adj A = C^T \) (cofactor transpose).
- Cramer’s Rule (for \( Ax = b \), det A ≠ 0): \( x_i = \frac{\det A_i}{\det A} \) (A_i replaces i-th column with b).
3. Linear Systems (Ax = b)
-
Gaussian Elimination: Row reduce [A b] to REF/RREF. - REF: Pivots (leading 1s) staircase down-right; zeros below pivots.
- Back-substitution for unique solution.
- Solutions:
- Unique: rank A = n (full column rank), nullspace {0}.
-
Infinite: rank A = rank [A b] < n (free variables). -
None: rank A < rank [A b].
- Complete Solution: Particular solution + nullspace basis (homogeneous solutions).
- LU Decomposition (no pivoting): A = LU (L lower unit triangular, U upper); solve Ly = b, Ux = y.
- Least Squares (overdetermined): \( \hat{x} = (A^T A)^{-1} A^T b \) (if full rank).
4. Vector Spaces & Subspaces
- Vector Space: Closed under addition/scalar mult.; axioms (e.g., 0 vector, inverses).
- Subspaces: Span of vectors; closed, contains 0.
- Column Space: Col(A) = span(columns of A); dim = rank A.
- Row Space: Row(A) = Col(A^T); dim = rank A.
-
Nullspace: Nul(A) = {x Ax = 0}; dim = n - rank A. - Left Nullspace: Nul(A^T).
- Linear Independence: c1 v1 + … + ck vk = 0 ⇒ all ci = 0.
- Basis: Lin. indep. spanning set.
- Dimension: # vectors in basis; dim Col(A) + dim Nul(A) = n (rank-nullity).
- Rank: # pivot columns = dim Col(A) = dim Row(A).
5. Linear Transformations
- Definition: T: V → W linear if T(u + v) = T u + T v, T(cu) = c T u.
- Matrix Rep.: [T] wrt bases = A where T(x) = A x (std. basis).
- Kernel: Ker T = Nul(A); Image: Im T = Col(A).
- Isomorphism: 1-1 onto (invertible matrix).
- Rank-Nullity: dim Ker T + dim Im T = dim V.
6. Eigenvalues & Eigenvectors
- Definition: A v = λ v (v ≠ 0 eigenvector, λ eigenvalue).
- Characteristic Eq.: det(A - λ I) = 0; roots λi (algebraic multiplicity).
- Eigenvectors: Solve (A - λ I) v = 0; geometric mult. = dim eigenspace.
- Diagonalizable: n lin. indep. eigenvectors ⇒ A = X Λ X^{-1} (Λ diag(λi), X = [v1 … vn]).
- Symmetric A: Always diagonalizable; orthogonal eigenvectors (A = Q Λ Q^T, Q orthogonal).
- Trace: tr A = ∑ λi.
- Det: det A = ∏ λi.
- Similar Matrices: A ~ B if A = P B P^{-1}; same eigenvalues, trace, det.
7. Inner Products & Quadratic Forms
-
Inner Product: <u, v> = u^T v (Euclidean); v = √<v,v>. - Orthogonal: <u,v> = 0; Orthonormal basis: <ei, ej> = δij.
- Gram-Schmidt: Orthogonalize basis {v1,…,vn} → {u1,…,un}.
- u1 = v1; uk = vk - proj_{span(u1..u_{k-1})} vk; proj_w v = (<v,w>/<w,w>) w.
- Quadratic Form: q(x) = x^T A x (A symmetric).
- Positive Definite: q(x) > 0 for x ≠ 0 (all λi > 0).
- Diagonalize: q(x) = ∑ λi yi^2 (y = Q^T x).
Quick Tips
- Compute rank: Row reduce to find # pivots.
- Check diagonalizable: Geometric mult. = algebraic mult. for each λ.
- For exams: Practice row reduction, det expansion, eigenproblems on 2x2/3x3 matrices.
- Common Errors: Forgetting multiplicity; sign in cofactors; non-commutative mult.
Good luck tomorrow—focus on understanding over memorizing!
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