Linear Algebra and Matrices
Explore matrices, linear systems, vectors, vector spaces, eigenvalues and linear transformations, including Gram-Schmidt, LU, QR, SVD and the pseudoinverse. Exact calculations include step-by-step working whenever available.
Version 2.0 Beta. Core features have been tested: numerical approximations are clearly distinguished from exact results. Always verify results before using them for important work.
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INTERACTIVE LAB
Set up the calculation
EXACT RESULT
Result
Step-by-step solution
For Gaussian elimination, I show the pivot, the multiplier calculation, substitution into the formula and the resulting matrix.
Pivot used Eliminated entry
Interpreting the result · Determinant, rank and invertibility
For a square matrix, det(A) ≠ 0 means full rank and the existence of an inverse. Rectangular matrices have a rank, but no overall determinant. Swapping two rows changes the sign of the determinant; adding a multiple of one row to another leaves it unchanged.
INTERACTIVE LAB
Solve a linear system
EXACT SOLUTION
Solutions of the system
Step-by-step solution
The steps follow the selected method. For Rouché–Capelli, I distinguish the coefficient matrix A, augmented matrix [A | b], their ranks and the system classification, without solving for the unknowns.
Interpreting the result · Consistency does not mean the solutions have been computed
Compare rank(A), rank([A | b]) and the number n of unknowns. Different ranks: no solution. Both ranks equal to n: one solution. Equal ranks less than n: infinitely many solutions. Rouché–Capelli classifies the system; Gaussian and Gauss-Jordan elimination can also determine the unknowns.
INTERACTIVE LAB
Analysis of a parameter-dependent system
Enter the real parameter a in the coefficients or the constant terms. The program identifies critical values and explains when the system has a unique solution, infinitely many solutions or no solution.
EXACT PARAMETER ANALYSIS
Case analysis
Step-by-step solution
I distinguish the coefficient and augmented matrices, examine minors to find values at which ranks may change, and classify each case. When possible, I show separately Cramer’s formulas.
Interpreting the result · Critical parameter values must be checked, not discarded
First I find parameter values at which the rank may change. Then I substitute each value and independently recompute rank(A) and rank([A | b]). Division by an expression that vanishes at a critical value is not valid. Irrational roots are classified symbolically.
INTERACTIVE LAB
Vector operations
Enter vector components and choose an operation. The lab shows the result, component calculations and geometric interpretation when useful.
CALCULATION RESULT
Result
Step-by-step solution
Each step explains the rule used and shows calculations for individual components.
Interpreting the result · Linear dependence and angles
Independent vectors: rank equals the number of vectors. Dependent vectors: at least one can be expressed as a combination of the others. A zero dot product indicates orthogonality, but the angle is undefined if either vector is zero. The cross product requires three components.
INTERACTIVE LAB
Vector spaces and linear maps
The columns of A represent vectors. Study their span, check whether another vector lies in it, or compute the null space of the associated linear map.
EXACT RESULT
Result
Step-by-step solution
I show Gauss-Jordan steps, the pivot positions and explanations for the selected operation.
Interpreting the result · Null-space and image bases
A basis of the image uses pivot columns from the ORIGINAL matrix, not the reduced matrix. A basis of the null space is obtained by solving Ax=0 and assigning parameters to free variables. For an m×n matrix, rank(A) + nullity(A) = n.
INTERACTIVE LAB
Eigenvalues, eigenvectors and diagonalization
Given a square matrix A, compute its characteristic polynomial, find eigenvalues, solve (A − λI)v = 0 and determine whether it is diagonalizable.
SPECTRAL ANALYSIS
Eigenvalues
Step-by-step solution
Find the characteristic polynomial, eigenvalues and, when possible, bases of eigenspaces. To diagonalize, I verify the identity A = PDP⁻¹.
Interpreting the result · Repeated eigenvalues and number fields
For each eigenvalue λ, solve (A−λI)v=0. Geometric multiplicity is the dimension of this null space, not the number of times λ appears as a root of the characteristic polynomial. A matrix is diagonalizable if the sum of eigenspace dimensions equals n, over the specified field ℝ or ℂ.
INTERACTIVE LAB
Linear transformations in the plane
A 2×2 matrix transforms points, vectors and shapes in the plane. See how the unit square, basis axes and area change. You can drag the vector in the graph.
EXACT RESULT
Plane transformation
Step-by-step solution
I show how the matrix acts on basis vectors, how to calculate A·u and why the determinant represents oriented area.
Interpreting the result · Area and orientation
In 2D, |det(A)| is the area scale factor. If det(A)>0, orientation is preserved; if det(A)<0, it is reversed. If det(A)=0, the square collapses and orientation is undefined. Rotations and reflections map the square to a congruent square.
INTERACTIVE LAB
Matrix factorizations, SVD and least squares
Choose a method and enter a matrix. Gram-Schmidt, LU, QR and full-column-rank least squares use exact fractions. SVD and the pseudoinverse use numerical approximations with error checks.
EXACT CALCULATION
Result
Step-by-step solution
I show transformations, multipliers, projections and checks. Radicals in the QR factorization remain exact.
Interpreting the result · Assumptions and numerical precision
Pivoted LU verifies PA=LU; a singular matrix may still admit an LU factorization, but not a unique solution to Ax=b. Reduced QR and exact least squares require full column rank. SVD, pseudoinverse and projectors are numerical; exact fractions are displayed only when certified by the Moore–Penrose identities.
How does the calculator work?
Exact calculations. Algebraic operations use fractions: a value such as 1/3 remains a fraction. Norms and distances are shown with radicals when necessary; angles are approximated in degrees.
Matrices and systems. Determinants can be evaluated using Gaussian elimination, Sarrus’s rule or Laplace expansion. Linear systems support Gauss, Gauss-Jordan, Cramer and Rouché–Capelli; the last classifies systems by comparing ranks without calculating the solution. Parameter-dependent systems identify critical parameter values.
Vectors. You can add vectors and calculate products, norms, distances, projections, angles and linear independence. Operations show individual component calculations, with geometric diagrams in the plane.
Vector Spaces. Find bases, dimensions, membership, null spaces and images, and verify the rank–nullity theorem using exact Gauss-Jordan steps.
2D Transformations. The SVG graph displays squares, grids, basis vectors and draggable vectors before and after transformation, illustrating the geometric significance of the determinant and invertibility.
Eigenvalues. Study 2×2 and 3×3 matrices, characteristic polynomials, eigenspaces, multiplicities and diagonalization. Results are exact for rational roots and supported quadratic radicals; non-factorable cubics are reported explicitly as approximations.
Advanced Methods. Gram-Schmidt, pivoted LU, reduced QR and full-column-rank least squares use exact fractions. Compact SVD and the Moore–Penrose pseudoinverse also support orthogonal projectors, rank-deficient least squares and underdetermined systems. SVD is numerical; rational pseudoinverses are labelled exact only after certification. A rotatable 3D diagram shows projections and residuals onto subspaces of R³.