Date: 27 April 2017

These are mainly notes to myself (i.e., Siddhartha Gadgil) but are shared in case they are useful.

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,...

The Optimization problem

Consider a collection of points in the plane, or more generally in $R^n$. In many cases, we want to find the line that is closest to this collection of points - or more generally the plane, or affine space of a fixed dimension. Two common reasons for this are: