Linear and multilinear algebra

Linear algebra studies mathematical objects like vectors, matrices, and linear transformations, which preserve addition and scalar multiplication. Multilinear algebra generalizes these ideas to functions that are linear in more than one variable, such as the dot product or determinants.

Important properties

• A linear transformation $T$ satisfies T(aoldsymbol{v} + boldsymbol{w}) = aT(oldsymbol{v}) + bT(oldsymbol{w}) for all vectors $\boldsymbol{v}, \boldsymbol{w}$ and scalars $a, b$.

• Matrices can represent linear transformations between vector spaces.

• The determinant and trace are important functions associated with matrices.

• Multilinear maps are functions that are linear in each argument separately.