Summary of contents: algebraic fields; vector spaces over an arbitrary field; vector (or linear) subspaces; linear maps; linear isomorphisms and isomorphic vector spaces; Hom-spaces; endomorphisms and automorphisms; dual vector space and linear functionals (covectors/one-forms); bilinear and multilinear maps; tensors and tensor product; examples; equivalence of endomorphisms and (1,1)-tensors; Hamel bases; linear independence and spanning set; dimension; double dual; dual bases and isomorphism of a vector space and its dual in finite dimensions; components of vectors and tensors; change of basis formulas; Einstein's summation convention and examples; column and row vectors and matrices; change of components under a change of basis; bilinear forms; permutations, symmetric group, transpositions, and signature of a transposition; totally anti-symmetric tensor; n-forms; volume-form and volume; determinant of an endomorphism.
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