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Quizzes > Mathematics

Linear Algebra Quiz: Applied Practice Questions

15 focused linear algebra practice problems with instant results.

Editorial: Review CompletedCreated By: Iskaltes IskaltUpdated Aug 23, 2025
Difficulty: Moderate
Questions: 15
Study OutcomesAdditional Reading
3D voxel art representing the Applied Linear Algebra course

This applied linear algebra quiz helps you check your understanding and build speed with matrix operations, linear systems, vector spaces, and eigenvalues. Tackle 15 quick questions with instant feedback, then use your results to focus your study; for extra practice, try the matrices quiz, an algebra quiz, or a calculus quiz.

For A in R^{m×n} and x in R^n, what is the dimension of the product Ax?
R^m
R^n
R^{m+n}
R^{m+n}
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Which property do orthogonal matrices Q satisfy in R^n?
Q^T Q = Q Q^T = 2I
Q is necessarily upper triangular
det(Q) = -2
Q^T Q = I and ||Qx||_2 = ||x||_2 for all x
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Which expression equals the Frobenius norm of A in R^{m×n}?
||A||_F = sqrt(tr(A^T A))
||A||_F = largest singular value of A
||A||_F = sum of eigenvalues of A
||A||_F = sqrt(det(A^T A))
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Which inequality is always true for ranks of matrices A and B with conformable dimensions?
rank(AB) = rank(A) + rank(B)
rank(AB) ≥ max(rank(A), rank(B))
rank(AB) = rank(A) = rank(B)
rank(AB) ≤ min(rank(A), rank(B))
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If the n column vectors of a square matrix B are a basis of R^n, how can a vector x be written using coordinates c in that basis?
x = B c
x = c^T B
x = B^{-1} c
x = B^T c
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Which statement is correct for a square matrix A regarding invertibility?
A is invertible iff rank(A) = n
A is invertible iff det(A) = 0
A is invertible iff A has at least one zero eigenvalue
A is invertible iff its columns are linearly dependent
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Which condition guarantees that an n x n matrix A is diagonalizable (over R or C as appropriate)?
A has rank less than n
A has n distinct eigenvalues
A is upper triangular
A is singular
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Which property characterizes an orthogonal projector P onto a subspace of R^n?
P^2 = I and P^T = -P
P^2 = P and P^T = P
P is orthogonal if and only if P^2 = 0
P^2 = P and P is upper triangular
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What is the closed-form ridge regression estimator for minimizing ||Ax - b||_2^2 + λ||x||_2^2 (λ > 0)?
x = (A A^T + λ I)^{-1} b
x = A^+ b regardless of λ
x = (A^T A + λ I)^{-1} A^T b
x = A^T (A A^T)^{-1} b
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For A in R^{m×n}, the rank-nullity theorem states n = rank(A) + nullity(A).
False
True
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Which statement describes the Eckart-Young theorem for SVD-based approximation?
PCA components cannot be obtained from SVD
Truncating the SVD to rank k minimizes the spectral radius
Truncating the SVD to rank k gives the best rank-k approximation in Frobenius norm
Any rank-k factorization has the same Frobenius error
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Gershgorin's circle theorem guarantees which of the following for a square matrix A?
Eigenvectors lie at disk centers
A is diagonalizable if all Gershgorin disks overlap
All eigenvalues of A lie in the union of its Gershgorin disks
The determinant equals the product of disk radii
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What is a correct property of the Rayleigh quotient R_A(x) = (x^T A x)/(x^T x) for symmetric A?
R_A(x) is independent of x
R_A(x) equals the trace of A
If x is an eigenvector, R_A(x) equals its eigenvalue
R_A(x) is always zero
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Which formula defines a Householder reflector H that maps a given vector to a coordinate axis direction?
H = Q R with Q orthogonal and R upper triangular
H = I - 2 uu^T/(u^T u) for a suitable nonzero u
H = 2I - uu^T
H = I + uu^T
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Sherman-Morrison gives (A + u v^T)^{-1} when A^{-1} exists and 1 + v^T A^{-1} u ≠ 0. Which is correct?
(A + u v^T)^{-1} = (I + u v^T) A^{-1}
(A + u v^T)^{-1} = A^{-1} - (A^{-1} u v^T A^{-1})/(1 + v^T A^{-1} u)
(A + u v^T)^{-1} = A^{-1} + u v^T
(A + u v^T)^{-1} = A - u v^T
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Schur decomposition states that any complex square matrix A can be written as:
A = L U with L lower bidiagonal and U upper bidiagonal
A = U Σ V^T with U,V unitary and Σ diagonal
A = Q T Q^*, with Q unitary and T upper triangular
A = P D P^{-1} with D diagonal for all A
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In PageRank with damping factor α in (0,1), the PageRank vector r solves which linear system (with teleport vector v)?
(I + α P) r = v
(P - I) r = (1 - α) v
P r = r regardless of α
(I - α P^T) r = (1 - α) v
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Which identity holds for Kronecker products when dimensions conform?
(A ⊗ B)^{-1} = A^{-1} + B^{-1}
(A ⊗ B) + (C ⊗ D) = (A + C) ⊗ (B + D) always
det(A ⊗ B) = det(A) + det(B)
(A ⊗ B)(C ⊗ D) = (A C) ⊗ (B D)
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Which identity is the Woodbury matrix identity for invertible A and suitable U,C,V?
(A + U C V)^{-1} = (I + U C V) A^{-1}
(A + U C V)^{-1} = A^{-1} - A^{-1} U (C^{-1} + V A^{-1} U)^{-1} V A^{-1}
(A + U C V)^{-1} = A - U C V
(A + U C V)^{-1} = A^{-1} + U C V
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What governs the asymptotic convergence rate of the power method for a diagonalizable A with |λ1| > |λ2|?
The condition number of A in 1-norm
The determinant of A
The ratio |λ2/λ1|
The trace of A
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Study Outcomes

  1. Apply matrix operations and determinants to solve systems of linear equations.
  2. Analyze vector spaces and linear transformations in engineering applications.
  3. Evaluate eigenvalues and eigenvectors to assess system stability.
  4. Interpret inner products, norms, and orthogonality within various problem contexts.
  5. Solve linear dynamical systems to model equilibrium behaviors.

Applied Linear Algebra Additional Reading

Here are some top-notch resources to supercharge your linear algebra journey:

  1. Dive into a comprehensive course featuring video lectures, assignments, and exams, all guided by the renowned Prof. Gilbert Strang. Perfect for building a solid foundation in linear algebra.
  2. This self-paced course offers lecture videos, problem sets, and exams with solutions, making it ideal for independent learners aiming to master linear algebra concepts.
  3. Explore a variety of online courses from top universities, covering topics from basic matrix operations to advanced applications in machine learning and beyond.
  4. This free study guide provides clear explanations and exercises on key topics like vector spaces, linear equations, and eigenvalues, complete with answers to odd-numbered exercises for self-assessment.
  5. For those seeking a more rigorous treatment, this course delves into the theoretical aspects of linear algebra, emphasizing proofs and deeper understanding of concepts.
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