Linear Algebra

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

Compute det(λI − A) with MiniWebtool’s Characteristic Polynomial Calculator. A square matrix from 2×2 to 6×6 produces expanded and factored polynomial forms, eigenvalues with multiplicities, and worked matrix steps.

Factor a symmetric positive-definite matrix as A = LLᵀ with MiniWebtool’s Cholesky Decomposition Calculator. The lower-triangular factor is built step by step, then multiplied by its transpose to verify reconstruction of the input.

Find a matrix’s range with MiniWebtool’s Column Space Calculator. Row reduction identifies pivot columns, then the corresponding original columns form a basis; rank, nullity, and step-by-step operations explain the result.

Solve a two- or three-variable linear system using determinants in MiniWebtool’s Cramer's Rule Calculator. Coefficients and constants produce column-replacement steps and determinant ratios when the system has a unique solution.

MiniWebtool’s Cross Product Calculator finds the vector perpendicular to two 3D vectors. Components yield the cross product, magnitude, and related angle, with determinant expansion and a rotatable diagram verifying perpendicularity.

MiniWebtool’s Determinant Calculator evaluates a square matrix and explains the cofactor expansion behind the result. Enter the matrix to inspect the determinant and what it indicates about properties such as singularity.

MiniWebtool’s Dot Product Calculator sums paired vector-component products in 2D, 3D, or higher dimensions. The scalar result is accompanied by magnitudes, angles, and projections where defined, with steps and a vector diagram.

Find a matrix’s invariant directions with MiniWebtool’s Eigenvalue and Eigenvector Calculator. A 2×2 or 3×3 matrix yields eigenvalues and associated eigenvectors, with characteristic-polynomial steps and transformation geometry.

Follow each projection in the Gram-Schmidt process with MiniWebtool’s Gram-Schmidt Calculator. Supply linearly independent vectors to obtain orthogonal and orthonormal bases, with intermediate calculations and checks showing how the resulting vectors relate.

MiniWebtool’s Jacobian Matrix Calculator arranges the partial derivatives of a vector-valued function into a matrix. Enter up to six components and variables for worked steps, with determinant and eigenvalue calculations where applicable.

Some matrices cannot be diagonalized, but can be expressed through Jordan blocks. MiniWebtool’s Jordan Normal Form Calculator calculates J and a transition matrix P, explaining generalized eigenvector chains and displaying the block structure.

Work through common matrix operations in MiniWebtool’s Matrix Calculator. Addition, subtraction, multiplication, transpose, determinant, and inverse calculations come with structured matrix displays and steps explaining how the entries are obtained.

Diagonalize a square matrix with MiniWebtool’s Matrix Diagonalization Calculator by computing eigenvalues, eigenvectors and the decomposition A = PDP⁻¹. It supports 2×2 to 5×5 matrices with step-by-step solutions, the characteristic polynomial, multiplicity analysis and interactive visualization.

Use a matrix exponential to examine a linear differential system. MiniWebtool’s Matrix Exponential Calculator evaluates e^(At) and can apply it to an initial vector, with numerical-method details and a phase-plane flow visualization.

Find the inverse of a 2×2 through 6×6 square matrix with MiniWebtool’s Matrix Inverse Calculator. Gauss-Jordan working and the determinant explain invertibility, while multiplication by the result checks for the identity matrix.

MiniWebtool’s Matrix LU Decomposition Calculator factors a square matrix up to 8×8 using partial pivoting. The lower, upper, and permutation matrices show how elimination and row exchanges combine to reproduce the original system.

Follow each row-by-column dot product in MiniWebtool’s Matrix Multiplication Calculator. Enter two compatible matrices up to 5×5 to see the product and the individual calculations used to fill its entries.

MiniWebtool’s Matrix Power Calculator raises a square matrix to a selected integer exponent. Intermediate calculations and matrix-property context help explain repeated multiplication, with negative powers requiring an invertible matrix.

Determine the number of independent rows or columns with MiniWebtool’s Matrix Rank Calculator. It handles matrices up to 10×10 and shows row-echelon working alongside the column-space and null-space dimensions.

The trace is the sum along a square matrix’s main diagonal. MiniWebtool’s Matrix Trace Calculator highlights those entries for matrices up to 10×10 and compares the total with the eigenvalue sum.

MiniWebtool’s Null Space Calculator finds a basis for all vectors satisfying Ax = 0. Exact-fraction row reduction exposes free variables, with nullity and a rank-nullity check connecting the solution set to the matrix’s dimensions.

Split a rational expression into simpler terms using MiniWebtool’s Partial Fraction Decomposition Calculator. Coefficient calculations and a visual breakdown explain how the separate fractions combine to reproduce the original function.

MiniWebtool’s QR Decomposition Calculator uses Gram-Schmidt orthogonalization to factor supported 2×2 through 5×5 matrices into Q and R. Animated steps explain the projections, and an orthogonality check compares QᵀQ with the identity matrix.

Follow matrix reduction in MiniWebtool’s RREF Calculator (Row Echelon Form). Each pivot, row swap, and elimination leads to reduced row echelon form, with rank, nullity, and solution information for an augmented matrix.

MiniWebtool’s Singular Value Decomposition (SVD) Calculator factors a real matrix up to 10×10 into U, Sigma, and V transpose, with rank and condition number for assessing its numerical structure.

Compute a Kronecker product with MiniWebtool’s Tensor Product Calculator. Two rectangular matrices produce an exact-fraction result, with block-by-block visualization and copyable output explaining how each entry expands the second matrix.

MiniWebtool’s Unit Vector Calculator normalizes a nonzero vector by its magnitude. Enter two, three, or up to fifty components to see the unit vector and applicable direction-angle information.

MiniWebtool’s Vector Calculator is a free online tool with step-by-step solutions for 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 with MiniWebtool’s Vector Magnitude Calculator. 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 with MiniWebtool’s Vector Projection Calculator. It supports 2D and 3D vectors with step-by-step formulas, an interactive diagram, and orthogonal decomposition.