Linear Algebra

Calculate the angle between two 2D or 3D vectors using the dot product formula cos(θ) = (a·b)/(|a||b|). Get step-by-step solutions, results in both degrees and radians, interactive vector diagram, and geometric interpretation.

Calculate the characteristic polynomial det(A − λI) of a square matrix. Supports 2×2 to 6×6 matrices with step-by-step cofactor expansion, eigenvalue extraction, coefficient analysis, and interactive polynomial visualization.

Decompose a symmetric positive-definite matrix into A = LLᵀ with animated step-by-step computation. See each element of the lower-triangular matrix L derived with full formulas, verify the result, and explore the factorization visually.

Find the column space and basis of any matrix using row reduction. See each row operation step by step with pivot column highlighting, rank, dimension, and interactive visualization for 2D/3D column spaces.

Solve systems of 2 or 3 linear equations using Cramer's rule. Enter coefficients, get step-by-step determinant calculations with animated matrix visualization, geometric interpretation graph, and the complete solution.

Calculate the cross product of two 3D vectors with the determinant formula. Get a step-by-step expansion, the perpendicular result vector, its magnitude as the parallelogram area, a direction check and a 3D visualization.

Calculate the determinant of any square matrix with detailed step-by-step cofactor expansion, matrix properties analysis, and visual solutions.

Calculate the dot product of two vectors in 2D, 3D or higher dimensions. Get the angle between them, magnitudes, scalar and vector projections, the geometric meaning and step-by-step formulas with an interactive diagram.

Calculate eigenvalues and eigenvectors of 2x2 and 3x3 matrices with detailed step-by-step solutions, characteristic polynomial derivation, interactive visualization, and matrix property analysis.

Orthonormalize a set of linearly independent vectors using the Gram-Schmidt process. Get step-by-step projections, orthogonal and orthonormal bases, verification of orthogonality, and an interactive vector visualization.

Calculate the Jacobian matrix of a vector-valued multivariable function. Enter the transformation components to get every partial derivative, the determinant, eigenvalues, worked steps and a grid deformation visualization.

Compute the Jordan canonical form of a square matrix plus the transition matrix P with P inverse A P = J. Handles defective, non-diagonalizable matrices via generalized eigenvectors, with kernel-chain analysis and a block diagram.

Calculate matrix operations with step-by-step solutions: addition, subtraction, multiplication, inverse, transpose, and determinant. Visual matrix display with detailed explanations.

Diagonalize a square matrix by computing eigenvalues, eigenvectors, and the decomposition A = PDP⁻¹. Supports 2×2 to 5×5 matrices with step-by-step solutions, characteristic polynomial, multiplicity analysis, and interactive visualization.

Compute the matrix exponential e^(At) and use it as the state transition matrix for linear ODE systems. Enter A, a time t and an optional initial vector to get x(t), Pade scaling details and an animated phase-plane flow diagram.

Calculate the inverse of a square matrix using Gauss-Jordan elimination with detailed step-by-step row operations. Supports 2×2 to 6×6 matrices with exact fractional arithmetic, determinant calculation, and A×A⁻¹=I verification.

Calculate LU decomposition of any square matrix with partial pivoting. Get lower triangular (L), upper triangular (U), and permutation (P) matrices with step-by-step Gaussian elimination and verification.

Multiply two matrices with detailed step-by-step element computation. See each dot product broken down, with color-coded row × column visualization. Supports matrices up to 5×5 with interactive dimension controls.

Calculate the power of a square matrix A raised to any integer exponent n. See each multiplication step animated, intermediate matrices A¹ through Aⁿ, determinant and trace properties, with MathJax formulas and interactive visualization.

Calculate the rank of any matrix using Gaussian elimination (row echelon form). Get step-by-step row reduction, pivot analysis, column space and null space dimensions, and visual heatmap. Supports matrices up to 10×10.

Calculate the trace of a square matrix (sum of diagonal elements), verify its equality with the sum of eigenvalues, explore trace properties, and visualize the diagonal with an interactive heatmap. Supports matrices up to 10×10.

Find the null space (kernel) of any matrix by solving Ax = 0 using Gaussian elimination. Get the basis vectors, nullity, step-by-step RREF reduction, and rank-nullity theorem verification with exact fractional arithmetic.

Decompose rational functions into partial fractions with detailed step-by-step solutions, coefficient analysis, and visual decomposition breakdown.

Decompose any matrix into an orthogonal Q and an upper triangular R with the Gram-Schmidt process. Handles 2x2 to 5x5 matrices with animated step-by-step orthogonalization and a check that Q transpose times Q equals the identity.

Reduce any matrix to reduced row echelon form with detailed step-by-step row operations. Enter your matrix and see every pivot, swap and elimination, then read off rank, nullity, pivot columns, free variables and solution type.

Calculate the singular value decomposition of any matrix, factoring A into U, Sigma and V transpose. Includes step-by-step solutions, an interactive 3D visualization, rank analysis, condition number and data compression uses.

Compute the tensor product, also called the Kronecker product, of two rectangular matrices with exact fraction arithmetic, block-by-block visualization, copyable results, and SEO-friendly linear algebra explanations.

Calculate the unit vector in the direction of any 2D, 3D or n-dimensional vector. Get the magnitude, each normalized component, direction angles, the step-by-step normalization and visual proof that the result has length 1.

Free online vector calculator with step-by-step solutions. Calculate dot product, cross product, magnitude, unit vector, angle between vectors, projection, and more with interactive 3D visualization.

Calculate the magnitude, or Euclidean norm, of a 2D, 3D or n-dimensional vector. Get a step-by-step computation showing each squared component, the sum under the radical and the final square root, with a vector diagram.

Calculate the vector projection and scalar projection of one vector onto another. Supports 2D and 3D vectors with step-by-step formulas, interactive diagram, and orthogonal decomposition.