Linear Algebra Practice Questions: Test Your Knowledge | LearnByTeaching.ai
Challenge your understanding of linear algebra with these 40 multiple-choice questions spanning vector spaces, matrix operations, eigenvalues, and inner product spaces. These questions test both computational skills and the geometric intuition that separates surface-level knowledge from true understanding.
40 questions total
Systems of Equations and Matrix Operations
Test your understanding of systems of equations and matrix operations.
What is the rank of the matrix [[1, 2, 3], [2, 4, 6], [0, 1, 1]]?
A homogeneous system Ax = 0 with 5 unknowns and 3 equations always has:
If A is a 3x3 matrix with det(A) = 4, what is det(2A)?
Which of the following is NOT an elementary row operation?
If AB = I, which statement is necessarily true?
The null space of a 4x6 matrix A with rank 3 has dimension:
What does it mean geometrically when a 2x2 matrix has determinant 0?
For the system Ax = b, if A is invertible, the unique solution is:
The column space of a matrix A is the same as:
If A is row equivalent to the identity matrix, then A is:
Vector Spaces and Subspaces
Test your understanding of vector spaces and subspaces.
Which of the following is NOT a vector space?
The dimension of the vector space of all 3x3 symmetric matrices is:
If W is a subspace of R^4 with dim(W) = 2, what is the dimension of its orthogonal complement W^perp?
Which of the following sets is a subspace of R^3?
If vectors v1, v2, v3 in R^3 are linearly independent, they form a:
The intersection of two subspaces of a vector space V is:
A linear transformation T: R^3 -> R^2 can be:
What is the dimension of the solution space of a homogeneous system with coefficient matrix of size 5x7 and rank 4?
The set of all solutions to a non-homogeneous system Ax = b (b ≠0) is:
If {v1, v2, v3, v4} spans R^3, then this set is:
Eigenvalues and Eigenvectors
Test your understanding of eigenvalues and eigenvectors.
If A is a 3x3 matrix with eigenvalues 1, 2, and 3, what is det(A)?
If λ is an eigenvalue of A, then λ^2 is an eigenvalue of:
A 4x4 matrix with characteristic polynomial (λ-2)^2(λ+1)(λ-3) has eigenvalue 2 with algebraic multiplicity:
A matrix A is diagonalizable if and only if:
The eigenvalues of a triangular matrix are:
If A has eigenvalue 0, then A is:
The trace of a matrix equals:
If A is a real symmetric matrix, its eigenvalues are:
For a 2x2 matrix with eigenvalues 3 and -1, what is the trace?
If A is diagonalizable with A = PDP^(-1), then A^k equals:
Inner Product Spaces and Orthogonality
Test your understanding of inner product spaces and orthogonality.
Two vectors u and v are orthogonal when:
The Gram-Schmidt process takes a set of linearly independent vectors and produces:
If Q is an orthogonal matrix, then Q^(-1) equals:
The projection of vector b onto the column space of A is given by:
If {u1, u2, u3} is an orthonormal basis for R^3 and v is any vector in R^3, then v equals:
The Cauchy-Schwarz inequality states that |u · v| is at most:
In the least squares solution to Ax = b, the residual vector b - Ax* is orthogonal to:
What is the norm of the vector (3, 4)?
The QR decomposition factors a matrix A into:
An orthogonal matrix preserves:
Scoring Guide
Total possible: 40
Study Recommendations
- Watch 3Blue1Brown's 'Essence of Linear Algebra' series for geometric intuition behind every concept
- Practice row reduction and eigenvalue computation by hand until the procedures are automatic
- For every theorem, construct a concrete 2x2 or 3x3 numerical example
- Connect abstract concepts to applications: PCA for data science, PageRank as an eigenvector problem
- Use the teach-back method — explain each concept aloud as if teaching a classmate
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