Dual of a vector space and the dual basis. The double dual of a vector space. Annihilator of a subset of a vector space. The transpose of a linear transformation. Real or complex inner product spaces: orthogonal complement of a linear subspace. Linear isometries and unitary operators: unitary and orthogonal matrices. Representation of a linear functional and adjoint of a linear operator. Self-adjoint operators: symmetric and hermitian operators. Spectral theorem for self-adjoint operators and for normal operators. Bilinear and sesquilinear forms: basis change, congruence, polar form. Scalar and hermitian products: existence of orthonormal bases. The Sylvester theorem: congruence criterion for symmetric matrices. Forms on inner product spaces: reduction to principal axes, simultaneous reduction of a pair of quadratic forms to a sum of squares.