Topics Covered

• Vector spaces, subspaces, algebra of subspaces, quotient spaces, linear combination of vectors, linear span, linear independence, basis and dimension, dimension of subspaces.

• Subspaces of Rn\mathbb{R}^nRn.
  Dimension of subspaces of Rn\mathbb{R}^nRn.
  Geometric significance of subspace up to R3\mathbb{R}^3R3.

• Four fundamental subspaces associated with a matrix.
  The dimension of the solution space of Ax=0Ax = 0Ax=0 and the rank of AAA.
  Full rank factorization, rank inequalities, Sylvester’s inequality.

• Linear transformations, null space, range, rank and nullity of a linear transformation.

• Matrix representation of a linear transformation, change of coordinate matrix.

• Algebra of linear transformations. Isomorphisms.
  Isomorphism theorems, invertibility and isomorphisms.

• Eigenvalues, eigenvectors and characteristic equation of a matrix (over C\mathbb{C}C).

• Cayley–Hamilton theorem and its use in finding the inverse of a matrix.