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.

Enter two 3D vector coordinates to calculate their cross product, perpendicular vector, and parallelogram area with step-by-step determinant expansion.

Enter a square matrix of up to 10 by 10 to get its determinant by cofactor expansion, with the steps and matrix properties.

Enter two vectors in 2D, 3D or up to 50 dimensions to get the dot product, the angle between them and the scalar and vector projections.

Enter a 2×2 or 3×3 matrix to get its eigenvalues and eigenvectors, the characteristic polynomial and the solution steps.

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.

Enter function components and variables (up to 6 each) to calculate the Jacobian matrix, determinant, eigenvalues, and worked steps.

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.

Enter a square matrix from 2x2 to 6x6. Get the inverse by Gauss-Jordan elimination, the determinant, and a check that A times A^-1 is I.

Enter a square matrix up to 8x8. Get the lower, upper, and permutation matrices from Gaussian elimination with partial pivoting.

Multiply two matrices up to 5x5 with step-by-step element computation and row-by-column breakdown.

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.

Enter a matrix up to 10x10. Get its rank from row echelon form, plus the column-space and null-space dimensions.

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.

Enter a matrix. Get a basis for the solutions of Ax = 0, the nullity, the row-reduction steps, and a rank-nullity check in exact fractions.

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.

Enter a matrix of at most 10 by 10. Get U, Sigma, and V transpose, plus the rank and the condition number.

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.

Enter a 2D, 3D, or longer vector, up to 50 dimensions, for its magnitude, unit vector, and direction angles.

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.