Title: Linear Algebra Essentials — LearnCoachAssist

Q: What is a scalar?
A: A single real (or complex) number, as opposed to a vector or matrix.

Q: What is a vector?
A: An ordered list of scalars; geometrically, a directed magnitude in space.

Q: What is a matrix?
A: A rectangular array of numbers arranged in rows and columns.

Q: What is a tensor?
A: A multi-dimensional array that generalizes scalars (0-D), vectors (1-D), and matrices (2-D) to higher orders.

Q: Define the dimension of a vector space.
A: The number of vectors in any basis of the space, written dim(V).

Q: What is a linear combination of vectors?
A: A sum of scalar multiples of the vectors: c₁v₁ + c₂v₂ + ... + cₙvₙ.

Q: Define linear independence.
A: A set of vectors is linearly independent if no vector in the set can be written as a linear combination of the others; equivalently, c₁v₁+...+cₙvₙ=0 implies all cᵢ=0.

Q: Define linear dependence.
A: A set of vectors is linearly dependent if there exist scalars, not all zero, such that their linear combination equals the zero vector.

Q: What is a basis of a vector space?
A: A linearly independent set of vectors that spans the entire space.

Q: What is the span of a set of vectors?
A: The set of all linear combinations of those vectors; it is itself a vector subspace.

Q: What is a subspace of a vector space?
A: A non-empty subset closed under vector addition and scalar multiplication, containing the zero vector.

Q: What is a column space of a matrix A?
A: The span of its columns, i.e. {Ax : x ∈ ℝⁿ}; a subspace of ℝᵐ.

Q: What is the null space (kernel) of a matrix A?
A: The set of all x with Ax = 0; a subspace of the domain.

Q: What is the nullity of a matrix?
A: The dimension of its null space.

Q: What is the rank of a matrix?
A: The dimension of its column space (equivalently row space).

Q: State the Rank-Nullity Theorem.
A: For an m×n matrix A: rank(A) + nullity(A) = n.

Q: What is a linear transformation?
A: A function T: V → W between vector spaces satisfying T(u+v)=T(u)+T(v) and T(cv)=cT(v).

Q: How is a linear transformation represented by a matrix?
A: By choosing bases for V and W, T(x) = Ax where A's columns are the images of the basis vectors of V.

Q: What is the determinant of a 2×2 matrix [[a,b],[c,d]]?
A: ad − bc.

Q: What does a zero determinant indicate?
A: The matrix is singular (non-invertible); its columns are linearly dependent and the linear map collapses dimension.

Q: What is the trace of a square matrix?
A: The sum of its diagonal entries; also equal to the sum of its eigenvalues.

Q: Define the transpose of a matrix.
A: The matrix obtained by swapping rows and columns; (Aᵀ)ᵢⱼ = Aⱼᵢ.

Q: Define the inverse of a square matrix A.
A: The unique matrix A⁻¹ such that AA⁻¹ = A⁻¹A = I, which exists iff det(A) ≠ 0.

Q: What is an identity matrix?
A: A square matrix with 1s on the diagonal and 0s elsewhere; the multiplicative identity.

Q: What is a diagonal matrix?
A: A square matrix whose off-diagonal entries are all zero.

Q: What is a triangular matrix?
A: A square matrix with all entries either above (upper) or below (lower) the diagonal equal to zero.

Q: What is a symmetric matrix?
A: A square matrix equal to its transpose: A = Aᵀ.

Q: What is a skew-symmetric (antisymmetric) matrix?
A: A square matrix with Aᵀ = −A, so its diagonal entries are zero.

Q: What is an orthogonal matrix?
A: A square matrix Q such that QᵀQ = QQᵀ = I; equivalently, its columns are orthonormal.

Q: What is a unitary matrix?
A: A complex square matrix U with U*U = UU* = I, where U* is the conjugate transpose.

Q: What is a positive definite matrix?
A: A symmetric matrix A such that xᵀAx > 0 for all non-zero real vectors x; all eigenvalues are positive.

Q: What is positive semidefinite?
A: A symmetric A with xᵀAx ≥ 0 for all x; eigenvalues are non-negative.

Q: Define an eigenvalue of a square matrix A.
A: A scalar λ such that Av = λv for some non-zero vector v.

Q: Define an eigenvector of A.
A: A non-zero vector v satisfying Av = λv for some eigenvalue λ.

Q: What is the characteristic polynomial of A?
A: det(A − λI); its roots are the eigenvalues of A.

Q: How do you compute eigenvalues of a 2×2 matrix [[a,b],[c,d]]?
A: Solve λ² − (a+d)λ + (ad−bc) = 0.

Q: What is diagonalization of a matrix?
A: Writing A = PDP⁻¹, where D is diagonal and P's columns are eigenvectors of A.

Q: When is a matrix diagonalizable?
A: Exactly when it has n linearly independent eigenvectors (i.e., its geometric multiplicity equals algebraic multiplicity for every eigenvalue).

Q: What is the geometric multiplicity of an eigenvalue?
A: The dimension of its eigenspace (null space of A − λI).

Q: What is algebraic multiplicity?
A: The number of times an eigenvalue appears as a root of the characteristic polynomial.

Q: What is the Cayley–Hamilton theorem?
A: Every square matrix satisfies its own characteristic polynomial: p(A) = 0.

Q: What is the minimal polynomial of a matrix?
A: The monic polynomial of least degree that annihilates the matrix.

Q: What is the spectral theorem for symmetric matrices?
A: Every real symmetric matrix can be orthogonally diagonalized: A = QDQᵀ with Q orthogonal and D real diagonal.

Q: State the singular value decomposition (SVD).
A: Any m×n matrix A can be written A = UΣVᵀ, with U, V orthogonal and Σ a diagonal matrix of non-negative singular values.

Q: What are singular values?
A: The square roots of the eigenvalues of AᵀA, arranged in non-increasing order on the diagonal of Σ.

Q: What does the rank of A equal in terms of the SVD?
A: The number of non-zero singular values.

Q: What is the Frobenius norm of a matrix?
A: The square root of the sum of squares of all entries: ‖A‖_F = √(Σᵢⱼ Aᵢⱼ²) = √(Σ σᵢ²).

Q: What is the L2 (spectral) norm of a matrix?
A: The largest singular value of A; the maximum factor by which A stretches any unit vector.

Q: Define the condition number of a square invertible matrix A.
A: κ(A) = ‖A‖·‖A⁻¹‖; for the 2-norm it is the ratio of largest to smallest singular value.

Q: Why is a large condition number problematic?
A: Small input perturbations can produce large output errors when solving Ax = b.

Q: What is the Moore–Penrose pseudoinverse A⁺?
A: The unique matrix satisfying AA⁺A = A, A⁺AA⁺ = A⁺, with AA⁺ and A⁺A Hermitian; for full-rank A it equals (AᵀA)⁻¹Aᵀ.

Q: What is the least-squares solution to Ax = b?
A: The vector x that minimizes ‖Ax − b‖²; given by the normal equations AᵀAx = Aᵀb, solved by x = (AᵀA)⁻¹Aᵀb.

Q: What are the normal equations?
A: AᵀAx = Aᵀb, whose solution minimizes ‖Ax − b‖².

Q: What is a projection matrix?
A: A square matrix P with P² = P; its image is the subspace onto which it projects.

Q: What is an orthogonal projection matrix?
A: A symmetric projection matrix (P = Pᵀ, P² = P); it projects onto its column space along the orthogonal complement.

Q: Define the inner (dot) product of two real vectors.
A: u·v = Σ uᵢvᵢ = uᵀv; zero iff u and v are orthogonal.

Q: What is the Cauchy–Schwarz inequality?
A: |u·v| ≤ ‖u‖·‖v‖, with equality iff u and v are linearly dependent.

Q: Define the Euclidean norm of a vector.
A: ‖x‖ = √(x·x) = √(Σ xᵢ²).

Q: What does it mean for vectors to be orthonormal?
A: Mutually orthogonal and each of unit length.

Q: What is the Gram–Schmidt process?
A: An algorithm that orthogonalizes a basis by successively subtracting projections and normalizing.

Q: What is a QR decomposition?
A: A = QR where Q has orthonormal columns and R is upper triangular.

Q: What is a Cholesky decomposition?
A: For a symmetric positive-definite A, A = LLᵀ with L lower triangular with positive diagonal entries.

Q: What is an LU decomposition?
A: A factorization A = LU with L lower triangular and U upper triangular, useful for solving linear systems.

Q: When does an LU decomposition without pivoting exist?
A: When every leading principal minor of A is non-zero (in particular, A is invertible).

Q: What is the difference between Ax = b and A⁻¹b?
A: Ax = b is the system; x = A⁻¹b is the solution when A is invertible.

Q: What is a row echelon form?
A: A form where each leading entry of a row is to the right of the leading entry of the row above, and rows of all zeros are at the bottom.

Q: What is the reduced row echelon form (RREF)?
A: Row echelon form where each leading 1 is the only non-zero entry in its column.

Q: What are elementary row operations?
A: (1) Swap two rows; (2) multiply a row by a non-zero scalar; (3) add a multiple of one row to another.

Q: How do row operations affect the solution set?
A: They do not change the solution set of Ax = b.

Q: What is the determinant of a triangular matrix?
A: The product of its diagonal entries.

Q: State the multiplicativity of the determinant.
A: det(AB) = det(A)·det(B) for square matrices of the same size.

Q: How does det(A) relate to det(Aᵀ)?
A: det(Aᵀ) = det(A).

Q: How does det(A) relate to det(A⁻¹)?
A: det(A⁻¹) = 1/det(A).

Q: What is the effect of swapping two rows of a matrix?
A: The determinant changes sign.

Q: What is the cofactor expansion of a determinant?
A: det(A) = Σⱼ (−1)^(i+j) aᵢⱼ Mᵢⱼ for any fixed row i, where Mᵢⱼ is the (i,j) minor.

Q: What is Cramer's rule?
A: A formula expressing each component of the solution to Ax = b as a ratio of determinants: xᵢ = det(Aᵢ)/det(A).

Q: What is a vector space?
A: A set V with addition and scalar multiplication satisfying associativity, commutativity, identity, inverses, and distributivity.

Q: Name the eight vector space axioms.
A: Closure under + and ·; commutativity and associativity of +; additive identity and inverses; multiplicative identity; distributivity of · over + and over scalar addition.

Q: What is the zero vector?
A: The unique additive identity in a vector space; satisfies v + 0 = v for all v.

Q: What is the additive inverse of a vector v?
A: The unique vector −v such that v + (−v) = 0.

Q: What is the difference u − v in a vector space?
A: u + (−v), the vector added to v gives u.

Q: What is a spanning set?
A: A set whose linear combinations cover the entire vector space.

Q: True or false: every vector space has a basis.
A: True; this is equivalent to the axiom of choice.

Q: What is the standard basis of ℝⁿ?
A: The set of n vectors e₁, …, eₙ with a 1 in one coordinate and 0 elsewhere.

Q: What is the change-of-basis matrix from basis B to basis C?
A: The matrix whose columns are the C-coordinates of the B basis vectors; it converts B-coordinates to C-coordinates.

Q: What does it mean for two matrices to be similar?
A: A = P⁻¹BP for some invertible P; they represent the same linear map in different bases.

Q: Do similar matrices have the same eigenvalues?
A: Yes; they share the same characteristic polynomial, trace, determinant, and rank.

Q: What is the Jordan canonical form?
A: An upper-triangular form with eigenvalues on the diagonal and 1s on the superdiagonal in Jordan blocks; every square matrix is similar to one.

Q: What is a Jordan block?
A: A square block with a single eigenvalue λ on the diagonal, 1s on the superdiagonal, and 0s elsewhere.

Q: What is the minimal polynomial of a diagonalizable matrix?
A: The product of distinct (x − λᵢ) factors over its eigenvalues λᵢ.

Q: What is the companion matrix of a polynomial p(x)=xⁿ+cₙ₋₁xⁿ⁻¹+...+c₀?
A: An n×n matrix with 1s on the subdiagonal, −cᵢ in the last row, and zeros elsewhere; its characteristic polynomial is p.

Q: What is a Hermitian (self-adjoint) matrix?
A: A complex matrix equal to its conjugate transpose: A = A*; real eigenvalues, orthogonally diagonalizable.

Q: What is a normal matrix?
A: A matrix commuting with its conjugate transpose: AA* = A*A; precisely the matrices unitarily diagonalizable.

Q: What is the Schur decomposition?
A: Any square matrix A = QTQ* with Q unitary and T upper triangular (diagonal entries are eigenvalues).

Q: What is the polar decomposition?
A: Any square invertible A = UP with U unitary/orthogonal and P positive definite.

Q: What is a stochastic matrix?
A: A square matrix with non-negative entries whose rows (or columns) sum to 1; used in Markov chains.

Q: What is the Perron–Frobenius theorem about?
A: A positive square matrix has a unique largest real eigenvalue (the Perron root) with a positive eigenvector; the spectral radius is a simple eigenvalue.

Q: Define the spectral radius of a matrix.
A: The maximum absolute value of its eigenvalues: ρ(A) = max |λᵢ|.

Q: What is Gershgorin's circle theorem?
A: Every eigenvalue lies in at least one disc centred at a diagonal entry aᵢᵢ with radius equal to the sum of absolute off-diagonal entries in row i.

Q: How do you test if a matrix is positive definite without computing eigenvalues?
A: By Sylvester's criterion: all leading principal minors must be positive.

Q: State the Cayley–Binet formula.
A: For A m×n, B n×m, det(AB) equals the sum of the determinants of all n×n submatrices formed by choosing n columns of A and the corresponding rows of B.

Q: What is a vector norm?
A: A function ‖·‖ assigning a non-negative length to each vector with ‖x‖=0 iff x=0, ‖cx‖=|c|·‖x‖, and ‖x+y‖≤‖x‖+‖y‖.

Q: What is the triangle inequality for norms?
A: ‖u + v‖ ≤ ‖u‖ + ‖v‖.

Q: What is Hölder's inequality for vectors?
A: Σ |uᵢvᵢ| ≤ ‖u‖_p · ‖v‖_q where 1/p + 1/q = 1.

Q: What is the relationship between matrix multiplication and composition of linear maps?
A: Matrix multiplication corresponds to function composition: the matrix of T∘S is the product of the matrices of T and S (in that order).

Q: What is the dimension of the solution space of Ax = 0 for an n-variable, rank-r system?
A: n − r (by rank–nullity).

Q: What is a Markov chain transition matrix?
A: A row-stochastic matrix P where entry Pᵢⱼ gives the probability of moving from state i to state j.

Q: How do you compute the steady state of a Markov chain?
A: Find the probability vector π with πP = π and Σπᵢ = 1; it is the eigenvector of Pᵀ with eigenvalue 1.

Q: What is the covariance matrix of a random vector X?
A: Σ = E[(X − μ)(X − μ)ᵀ]; symmetric and positive semidefinite.

Q: How does PCA use eigendecomposition?
A: The principal components are eigenvectors of the covariance matrix; eigenvalues give the variance explained along each component.

Q: Why are symmetric matrices important in applications?
A: They have real eigenvalues and orthogonal eigenvectors, arise naturally in quadratic forms, covariance matrices, and mechanics, and are numerically stable to diagonalize.

Q: What is a quadratic form?
A: A function Q(x) = xᵀAx for a symmetric matrix A; sign of Q classifies A as positive/negative definite or indefinite.

Q: What is the signature of a symmetric matrix?
A: The number of positive, negative, and zero eigenvalues (with multiplicity), e.g. (p, n−p, 0) for an n×n matrix.

Q: What is the inertia of a matrix?
A: The triple (n₊, n₋, n₀) of counts of positive, negative, and zero eigenvalues; preserved under congruence.

Q: What is Sylvester's law of inertia?
A: Two real symmetric matrices are congruent (SᵀAS = B) iff they have the same inertia.

Q: What does it mean for matrices to be congruent?
A: A and B are congruent if B = SᵀAS for some invertible S; preserves definiteness but not eigenvalues.

Q: What is the matrix exponential exp(A)?
A: The infinite series Σₖ Aᵏ/k!, convergent for any square A; solves dX/dt = AX with X(0)=I.

Q: State the derivative of det(A(t)) when A is differentiable.
A: d/dt det(A) = det(A) · tr(A⁻¹ A').

Q: What is the Jacobian matrix of a vector-valued function f: ℝⁿ→ℝᵐ?
A: The m×n matrix of partial derivatives Jᵢⱼ = ∂fᵢ/∂xⱼ.

Q: What does the determinant of the Jacobian represent geometrically?
A: The local volume scaling factor of the transformation near a point.