Linear algebra

Interpret the entries of a vector in a named coordinate system.

Add two vectors of the same dimension.

Scale each coordinate of a vector by a scalar.

Compute the displacement between two coordinate vectors.

Construct a weighted sum of supplied vectors.

Compute the dot product of two equal-length vectors.

Compute the Euclidean length of a vector.

Normalize a nonzero vector to unit length.

Compute the Euclidean distance between two vectors.

Decide whether two vectors are orthogonal using their dot product.

Compute directional similarity for two nonzero vectors.

Interpret a supplied embedding ranking within its stated representation.

Compute the L1 norm of a vector.

Identify the row and column dimensions of a matrix.

Retrieve a matrix entry using its row and column indices.

Add matrices of equal shape.

Multiply each matrix entry by a scalar.

Compute a matrix-vector product from row dot products.

Express a matrix-vector product as a combination of its columns.

Compute the entries of a compatible matrix product.

Transpose a rectangular matrix.

Interpret the coordinate scaling performed by a diagonal matrix.

Construct an outer product from two vectors.

Encode a finite linear system as Ax=b.

Apply one solution-preserving elementary row operation.

Eliminate a variable using elementary row operations.

Identify pivot positions in row-echelon form.

Solve a triangular system by back substitution.

Reduce an echelon matrix to reduced row-echelon form.

Detect a contradictory row in an augmented system.

Parameterize the free variables of a consistent reduced system.

Determine whether a consistent system has a unique solution.

Decide whether a vector lies in the span of supplied vectors.

Check the defining closure conditions for a candidate real subspace.

Test whether a supplied vector list is linearly independent.

Verify a basis for a specified finite-dimensional subspace.

Determine subspace dimension from a basis.

Find a basis for the column space of a matrix.

Determine matrix rank from its pivots.

Check additivity and homogeneity for a candidate map.

Represent a composition of linear maps as an ordered matrix product.

Separate the linear part from the translation in an affine map.

Compute the orthogonal projection onto a nonzero vector's span.

Verify that a supplied basis is orthonormal.

Project a vector using a supplied orthonormal basis.

Identify the closest attainable output in an inconsistent linear system.

Use supplied reduced QR factors to solve a full-column-rank least-squares problem.

Form the normal equations for a linear least-squares problem.

Interpret the three transformations in a supplied SVD.

Infer rank from the nonzero singular values of a matrix.

Construct a rank-k approximation from supplied singular components.

Compute a matrix's Frobenius norm.

Compute truncated-SVD Frobenius error from discarded singular values.

Interpret the ratio of largest to smallest nonzero singular values for a full-rank square system.