Linear Algebra

Work with the structure of a vector space

• Given a subset of a vector space described by a condition, the learner can decide whether it is a subspace, prove closure for arbitrary elements when it is, and exhibit an explicit counterexample naming the failed condition when it is not.
• Given a map between vector spaces, the learner can decide whether it is linear and justify the verdict, construct its matrix from the images of a basis when it is, and determine the kernel and image with their dimensions, checking the result against rank–nullity.
• Given a spanning set or a described subspace, the learner can extract or construct a basis, state the dimension with justification, compute the coordinates of a vector with respect to a stated basis, and decide whether a candidate set is a basis using the counting shortcut where it applies.
• Given a square matrix, the learner can compute its determinant by cofactor expansion or by elimination, choosing the cheaper route and justifying the choice, and can state what the value implies about invertibility, dependence of the columns, and the volume scaling of the associated map.
• Given a square matrix, the learner can form and solve the characteristic polynomial, compute a basis for each eigenspace, compare geometric with algebraic multiplicity, and say whether a basis of eigenvectors exists, checking the result against the trace and determinant.
• Given a square matrix, the learner can decide whether it is diagonalizable over a stated field, construct $P$ and $D$ when it is with the columns and diagonal entries correctly paired, verify the factorisation, and use it to compute a power of the matrix.
• Given a basis of a subspace, the learner can produce an orthogonal basis by Gram–Schmidt, project a vector onto that subspace, verify that the residual is orthogonal to the subspace, and set up the normal equations for a least-squares problem.
• Given a small matrix, the learner can compute its singular values from $A^{\mathsf{T}}A$, construct the right and left singular vectors, verify the factorisation, read the rank and the four subspaces off it, and give the best low-rank approximation with its error.
• Given a quadratic form written in variables, the learner can produce its symmetric matrix, compute the eigenvalues, classify the definiteness, give the principal axes and the diagonalised form, and confirm the verdict by evaluating the form on specific vectors.
• Given a square matrix, the learner can decide whether it is invertible, compute the inverse by Gauss–Jordan elimination, verify it, express the reduction as a product of elementary matrices, apply the order-reversal rule to a product, and produce an LU factorisation.
• Given vectors in ${\mathbb{R}}^3$, the learner can compute a cross product and verify its perpendicularity, use its length as an area, compute a scalar triple product as a volume and as a coplanarity test, and apply the distance formulas for a point to a line and to a plane.
• Given complex numbers in rectangular or polar form, the learner can add, multiply and divide them, take conjugates and moduli, convert between forms, apply De Moivre's theorem to powers and roots, and explain why the non-real eigenvalues of a real matrix arrive in conjugate pairs.
• Given a set with two operations, the learner can read and write set notation, test the field axioms one at a time, decide whether the structure is a field, and when it is not, name the specific axiom that fails and produce a witness to the failure.
• Given a permutation in one-line or cycle notation, the learner can convert between the notations, count inversions, decompose into disjoint cycles and transpositions, determine the sign by either route, compose permutations, and use the sign in the Leibniz formula and in reading a permutation matrix's determinant.