What are the column space and null space?
Answer: Column space: all linear combinations of the columns (range of A). Null space: all x with Ax = 0.
Explanation
Null space is nontrivial exactly when columns are dependent.
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Linear Algebra: Vectors, Matrices, and Eigenvalues (University)
SubspacesApplicationsOrthogonalityEigenTransformationsVectorsSystemsMatrices
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Subspaces
What is a subspace?
Answer: A subset closed under addition and scalar multiplication and containing the zero vector.
Explanation
Lines and planes through the origin are subspaces of R³.
Applications
What does the singular value decomposition (SVD) express?
Answer: Any matrix as A = UΣVᵀ with orthogonal U, V and diagonal Σ of singular values.
Explanation
Used in compression, PCA, and pseudo-inverses.
What is a Markov matrix and its steady state?
Answer: A matrix with nonnegative columns summing to 1; the steady state is the eigenvector for eigenvalue 1.
Explanation
Models transitions between states.
Orthogonality
What are least squares solutions?
Answer: The x minimizing |Ax − b|, found from the normal equations AᵀAx = Aᵀb.
Explanation
Used for regression when the system has no exact solution.
What is the Gram-Schmidt process?
Answer: A method to turn a basis into an orthonormal basis by subtracting projections onto previous vectors and normalizing.
Explanation
Produces the Q in QR factorization.
What is an orthonormal set?
Answer: Vectors that are mutually orthogonal and each have length 1.
Explanation
Orthogonal matrices have orthonormal columns: QᵀQ = I.
Eigen
What is special about symmetric matrices?
Answer: They have real eigenvalues and orthogonal eigenvectors; they are orthogonally diagonalizable.
Explanation
Spectral theorem.
Why is diagonalization useful?
Answer: Aᵏ = PDᵏP⁻¹ makes powers easy to compute; used in Markov chains and differential equations.
Explanation
Dᵏ just raises each eigenvalue to k.
What does it mean to diagonalize a matrix?
Answer: Write A = PDP⁻¹ with D diagonal (eigenvalues) and P's columns the eigenvectors.
Explanation
Possible when there are n linearly independent eigenvectors.
How do you find eigenvalues?
Answer: Solve the characteristic equation det(A − λI) = 0.
Explanation
Then find eigenvectors from (A − λI)v = 0.
What are eigenvalues and eigenvectors?
Answer: For a square matrix A, a nonzero vector v with Av = λv; λ is the eigenvalue.
Explanation
Eigenvectors keep their direction under the transformation.
Transformations
What does the kernel of a transformation tell you?
Answer: The set of inputs mapped to zero; T is one-to-one exactly when the kernel is {0}.
Explanation
Same as the null space of its matrix.
What matrix rotates the plane by angle θ?
Answer: [[cosθ, −sinθ], [sinθ, cosθ]]
Explanation
Columns are the images of the standard basis vectors.
What is a linear transformation?
Answer: A map T with T(u + v) = T(u) + T(v) and T(cu) = cT(u).
Explanation
Every linear map from Rⁿ to Rᵐ is multiplication by an m×n matrix.
Vectors
What is a linear combination?
Answer: A sum of scalar multiples of vectors: c1·v1 + c2·v2 + … + cn·vn.
Explanation
The set of all linear combinations is the span.
What is the cross product and what does it produce?
Answer: For 3D vectors, u × v is a vector perpendicular to both, with magnitude |u||v|sinθ (the parallelogram area).
Explanation
Direction by the right-hand rule.
Dot product formula and what it tells you
Answer: u·v = Σ uᵢvᵢ = |u||v|cosθ; zero means the vectors are orthogonal.
Explanation
Projection of u onto v: (u·v / v·v) v.
What is a basis of a vector space?
Answer: A linearly independent set that spans the space.
Explanation
The number of vectors in a basis is the dimension.
When are vectors linearly independent?
Answer: When the only solution to c1·v1 + … + cn·vn = 0 is all ci = 0.
Explanation
Dependent means at least one is a combination of the others.
Systems
What is the rank of a matrix?
Answer: The number of pivot columns (dimension of the column space).
Explanation
Rank-nullity: rank + nullity = number of columns.
When does a linear system have no solution, one solution, or infinitely many?
Answer: No solution: a row like [0 0 … | b] with b ≠ 0. One: a pivot in every variable column. Infinitely many: free variables exist.
Explanation
Consistent systems have zero or more free variables.
What is reduced row echelon form?
Answer: Leading 1s in each nonzero row, each leading 1 is the only nonzero entry in its column, leading 1s move right as you go down, zero rows at the bottom.
Explanation
RREF is unique for a given matrix.
Matrices
What are elementary row operations?
Answer: Swap two rows, multiply a row by a nonzero scalar, add a multiple of one row to another.
Explanation
Used in Gaussian elimination.
What does a zero determinant mean geometrically?
Answer: The transformation collapses space onto a lower dimension (area or volume becomes zero).
Explanation
Columns are linearly dependent.
How is the determinant of a 2×2 matrix computed?
Answer: det [[a, b], [c, d]] = ad − bc.
Explanation
For larger matrices use cofactor expansion or row reduction.
When does a square matrix have an inverse?
Answer: When its determinant is nonzero (it is nonsingular); then AA⁻¹ = I.
Explanation
Inverse of a 2×2: (1/det) · [[d, −b], [−c, a]].
What is the identity matrix?
Answer: A square matrix with 1s on the diagonal and 0s elsewhere; AI = IA = A.
Explanation
The multiplicative identity.
What is the transpose and a symmetric matrix?
Answer: The transpose swaps rows and columns (Aᵀ); a matrix is symmetric if A = Aᵀ.
Explanation
(AB)ᵀ = BᵀAᵀ.
When can two matrices be multiplied, and what size is the result?
Answer: When the columns of A equal the rows of B; an m×n times n×p gives m×p.
Explanation
Matrix multiplication is not commutative.
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