Linear algebra

• How to Combine Transformation Matrices
  Combine transformation matrices by multiplying them in reverse application order for column vectors, including translations with homogeneous coordinates.
• Difference Between the Dimension and Rank of a Matrix
  Learn why matrix size is not the same as rank, how rank is defined as a space dimension, and how to calculate both with an example.
• How to Find the Dimension of the Kernel of a Square Matrix
  For an n × n matrix, the kernel has dimension n minus the matrix rank. Row-reduction gives the rank and therefore the nullity.
• How to Prove a Homogeneous System Has Infinitely Many Solutions
  For a homogeneous system with n unknowns and m equations over an infinite field, rank-nullity guarantees a nonzero solution and infinitely many scalar multiples.
• How to Compute the Semisimple Part of a Matrix
  Compute a matrix’s semisimple part by replacing each Jordan block with its eigenvalue times the identity, or by evaluating a suitable polynomial.
• How to Pare Down a Set to Form a Basis
  To pare down a finite spanning set, row-reduce its column matrix and keep the original vectors in the pivot-column positions.
• How to Find the Rank of a Matrix A with A² = A
  If A² = A, then A is idempotent. Learn why its rank can range from 0 to n, how to compute it from the entries, and why rank equals trace.
• What Is an Example of a Degenerate Bilinear Map?
  A bilinear map is degenerate when a nonzero vector pairs to zero with every vector; see matrix tests and concrete examples in its space.
• Examples of Additive-Only and Multiplicative-Only Functions
  See examples of functions that preserve sums but not products, or products but not sums, and learn when an additive example must be pathological.
• How Do You Find the Dimensions of Subspace Sums and Intersections?
  For finite-dimensional subspaces, compute dimensions of sums and intersections with the dimension formula, and see why unions usually fail to be subspaces.
• Do All Linear Transformations Map Zero to Zero?
  Every linear transformation sends the domain's zero vector to the codomain's zero vector, as follows directly from additivity; the reverse need not hold.
• What Is the Trace of a Linear Operator?
  The trace of a finite-dimensional linear operator is the sum of its matrix's diagonal entries, a value that does not depend on the chosen basis.
• How to Find a Gram Matrix Determinant Without Special Methods
  Find a Gram matrix determinant using ordinary determinant rules, with a worked 3×3 example and a formula for symmetric matrices.
• What Is the Determinant of a Scalar Times a Matrix?
  For an n by n matrix A and scalar c, det(cA) equals c^n det(A), not simply c det(A).
• Why Do Arithmetic-Progression Columns Give a Zero Determinant?
  If every column is an arithmetic progression, each lies in the span of two fixed vectors, so an n×n determinant is zero for n at least 3.
• How to Determine Whether a Set Spans ℝ²
  Check whether vectors span ℝ² by testing whether their matrix has rank 2; for two vectors, compute the determinant and check it is nonzero.
• How to Diagonalize a 3×3 Matrix
  To diagonalize a 3×3 matrix, find its eigenvalues and eigenvectors, check that three independent eigenvectors form P, and verify AP = PD.
• What Conditions Make the Product of Two Matrices Zero?
  For compatible matrices, AB = 0 exactly when the image of B lies in the kernel of A; this also implies a rank bound.
• Why Do Rotation Matrices About the y-Axis Have Different Signs?
  For column vectors in a right-handed system, a positive active rotation about y has a specific sign pattern; the opposite is its inverse.
• How to Calculate an Eigenspace
  Calculate an eigenspace by solving the homogeneous system (A − λI)x = 0, then express its solution set as a span or basis.
• Can One Eigenvalue Have Two Different Eigenvectors?
  Yes. An eigenvalue can have many eigenvectors: every nonzero vector in its eigenspace qualifies, including scalar multiples and, sometimes, independent vectors.
• How to Find a Basis for Each Eigenspace
  Find an eigenspace by solving (A − λI)x = 0. A basis consists of independent eigenvectors that span the solutions, excluding zero.
• What Is an Example of a Nonstandard Inner Product?
  A weighted dot product such as 2x₁y₁ + 3x₂y₂ is a valid inner product that differs from the standard one in the same coordinates.
• Examples of 3×3 Matrices With Common Properties
  Browse verified 3×3 matrix examples for symmetry, rank, invertibility, orthogonality, idempotence, nilpotence, and noncommutativity.
• How to Calculate a Determinant by First-Row Cofactor Expansion
  Expand along the first row by multiplying each entry by its signed minor. See the alternating signs, a worked 3×3 example, and common errors.
• How to Find the Determinant of a Skew-Symmetric Matrix
  A skew-symmetric matrix has determinant zero when its order is odd; for even order, use its Pfaffian. See the 4 × 4 formula and example.
• Direct Sum vs. Direct Product vs. Tensor Product
  A direct sum restricts infinite tuples to finite support, a direct product allows all tuples, and a tensor product combines spaces bilinearly.
• How to Prove an Element Is Nonzero in a Tensor Product
  Over a field, detect a nonzero tensor by applying linear functionals; over a ring, use a balanced map or compute the module relations.
• How to Find the Inverse of a Rigid Transformation
  Find the inverse of a rigid transformation using its orthogonal matrix and translation, and verify that the inverse also preserves distances.
• How to Write Least Squares in Matrix Form
  Express least squares as minimizing the residual norm ||Ax − b||, derive the normal equations, and see a fully worked linear-fit example.
• What Is the Relationship Between Eigendecomposition and SVD?
  The SVD is linked to the eigendecompositions of A* A and A A*: their eigenvectors give singular vectors and their eigenvalues are squared singular values.
• What Is the Tensor Double Dot Product?
  The tensor double dot product contracts matching indices: for two matrices, it sums entrywise products and equals the trace of A transpose times B.
• Understanding Euler’s Rotation Theorem
  Euler’s rotation theorem says every proper rotation in three dimensions has a fixed axis. Learn why, how to find that axis, and how to calculate the rotation angle.
• What Is the Intuitive Relationship Between SVD and PCA?
  For a centered data matrix, PCA directions are the right singular vectors of its SVD; squared singular values give component variances.
• What Is the Tensor Product of a Vector with Itself?
  The tensor product of a vector with itself is an element of V⊗V; in coordinates, it is the outer product with entries v_i v_j.
• When Is an Element of a Tensor Product Zero?
  For vector spaces over a field, a pure tensor is zero exactly when one factor is zero; sums and tensors over general rings need separate tests.
• Why Are Complex Numbers Numbers but 2D Vectors Usually Aren’t?
  Complex numbers are numbers because they have a consistent arithmetic structure. A 2D vector becomes one if you define suitable multiplication.
• Inverting a Gaussian (RBF) Kernel Matrix Without Numerical Trouble
  A Gaussian kernel matrix is positive definite for distinct points, so it is invertible, but it is often badly conditioned. Use a ridge term and a Cholesky solve instead.