### Overview Topics

#### Optimization

• For functions of one variable, application of calculus.
• Have first derivative and second derivative test.
• In higher dimension:
• to define first derivative: linear map (actually functional in this case).
• second derivative : also captureb by a linear map - but a square symmetric matrix.
• we can also have interesting constraints.
• even more generally, we have calculus of variations.

#### Linear Algebra

• Study linear maps on vector spaces, because:
• many maps are linear
• many functions are quadratic - these also can be expressed in linear algebra
• smooth functions are locally approximately linear, with the second term quadratic.
• By study, we typically mean properties or structures that do not change when we chang co-ordinates.
• for a vector space, there is just one, the dimension.
• for linear map, we consider different classes.
• domain and co-domain equal (and change co-ordinates together).
• co-domain real numbers (functionals).
• more structure on vector spaces to be preserved - e.g. lengths and angles.
• special classes of maps, e.g. symmetric matrices.
• One linear map can be described completely:
• Sylvester’s law of inertia
• Jordan canonical form

#### Local to global

• Differential calculus lets us study local properties of functions.
• To deduce from this global behaviour, we integrate and solve differential equations.