Section A
Section A contains short and basic questions that test fundamental concepts of linear algebra. Students need to write definitions, state theorems, and solve small problems related to vector spaces, linear transformations, and quadratic forms. The questions are direct and check basic theoretical understanding.
Example Questions from Section A:
Define basis of a vector space.
State rank-nullity theorem.
Define complete orthonormal set.
Give polarization identity.
Define linear functional with example.
Find dimension of a vector space.
Section B
Section B includes medium-length questions that require explanations, proofs, and problem solving related to vector spaces, eigen values, and linear transformations. Students must demonstrate conceptual understanding and apply mathematical methods.
Example Questions from Section B:
Define field with example.
Prove that a given set is a vector space.
Find eigenvalues and eigenvectors of a matrix.
State and prove Minkowski inequality.
Prove union of two subspaces is a subspace under certain conditions.
Section C
Section C contains long and advanced questions that require detailed proofs, calculations, and application of linear algebra concepts. Students must solve problems related to linear dependence, orthogonal basis, rank, and transformations. This section tests deep analytical skills.
Example Questions from Section C:
Prove vectors are linearly dependent.
Find rank of a linear transformation.
Construct orthogonal basis of a vector space.
Check whether vectors are eigenvectors of a matrix.
Find image of vectors under a linear transformation.
