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

- 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*.

- 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,...

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:

- the points represent values of a function, and we want to approximate this by a linear function $y = mx + c”
- the points are data in some high-dimensional space - say many numbers associated to countries - and we want to capture variation in as...

- Let $V$ be a vector space over a field $k$.
- A
*linear functional*$\lambda$ on $V$ is a linear transformation $\lambda: V \to k$. - Linear functionals form a vector space, called the
*dual space*$V^*$ of $V$.

- Assume now that $V$ is a finite-dimensional vector space over $\mathbb{R}$ equipped with an inner product $\langle\cdot,\cdot\rangle$.
- Let $w\in V$ be a fixed vector. Then we can define a linear functional $\lambda_w: V \to\mathbb{R}$ by $\lambda_w(v) = \langle v, w \rangle$ for an arbitrary vector $v\in V$.
- We thus obtain a linear...

- Let $V$ be a vector space over a field $k$ in which $2 \neq 0$ (say real or complex numbers).
- A
*Bilinear form*$B(x, y)$ is a function $B : V \times V \to k$ which is linear in $x$ and $y$. - We can associate to a Bilinear form $B$ a so called
*quadratic form*$Q : V \to k$ defined by $Q(x) = B(x, x)$. Note that by definition a Quadratic form is associated to a Bilinear form. - Given the Bilinear form $B$, we can define a
**symmetric**Bilinear form $\bar{B}(x, y) =...