Mathematics III - Foundations of Analysis - Summer 24

We shall deal with maps $f:{\mathbb R}^n\to{\mathbb R}^m$ (from one Euclidean space to another), and try to understand what the derivative of such a map is.

Here are some notions from linear algebra which will be used here.

• vectors and vector space
• linear combination of vectors and span
• linearly (in)depencce and basis
• dimension of a vector space, in particular, dim ${\mathbb R}^n$ is $n$
• linear transformation  $A$ from a vector space $X$ to another $Y$.  $A$ is called a linear operator if $X=Y$.
• injective, surjective and bijective linear operator and inverse
• set of all linear transformations from $X$ to $Y$: $L(X,Y)$ or $L(X)$ if $X=Y$
• Composition (product) of two linear transformations $A:X\to Y$ and $B\to Y\to Z$ is $BA:X\to Z$.

Remark: $AB\neq BA$ even if $A,B\in L(X)$.

Theorem.  A linear operator $A$ on a finite dimensional vector space is one-to-one if and only if the range of $A$ is all of $X$.

Theorem.  Assume that the invertible linear operator $A$ of ${\mathbb R}^n$ satisfies the inequality $||B-A|| ||A^{-1}||$, for $B\in L({\mathbb R}^n)$.  Then $B$ is invertible.

Theorem. Assume ${\mathbf x}=\{x_1,...,x_n\}$ and ${\mathbf y}=\{y_1,...,y_m\}$ are bases of vector space $X$ and $Y$, respectively.  Then every $A\in L(X,Y)$ can be represented by a matrix $[A]$ in the sense that $[Y]_{\mathbf y}=[A][X]_{\mathbf x}$.

Here is perhaps a new notion: Norm of $A$: $||A||=\sup\{|Ax|: ||x||\le 1, \forall x\in{\mathbb R}^n\}$.

Theorem. The norm of $A\in L({\mathbb R}^n, {\mathbb R}^m)$ is finite and unifromly continuous, and for

||A+B||\le ||A||+||B||$, ||cA||=|c|||A||$ for scalar $c$, $A,B\in L({\mathbb R}^n, {\mathbb R}^m)$ ;

||BA||\le ||B|||A||, for $A\in L({\mathbb R}^n, {\mathbb R}^m)$, $B\in L({\mathbb R}^m, {\mathbb R}^k)$

With $||\cdot||$ defined as distance function $L({\mathbb R}^n, {\mathbb R}^m)$ is a metric space.  _◊

By Cauchy-Schwarz inequality, $\displaystyle ||A||\le(\sum_{i,j}a_{ij})^2$.

Proposition. If $S$ is a metric space, $a_{ij}$ ($i=1,...,m$,$j=1,...,n$) are real continuous functions on $S$, and if for each $p\in S$, $A_p\in L({\mathbb R}^n,{\mathbb R}^m)$ with matrix $(a_{ij})$, then the mapping $p\to A_p$ is a continuous mapping of $S$ into$L({\mathbb R}^n,{\mathbb R}^m)$.

How to compute the norm of $A$? For example what is $||A||$ if $A=\begin{pmatrix}1&2\\0&1\end{pmatrix}$?  From the above inequality we  get an upper bound for the norm:  $\sqrt{6}$.   Here is a computational formula for a norm of square matrix.

Theorem. (Peano) If $A$ is an $n\times n$ symmetric matrix, then all eigenvalues are real and

$\displaystyle||A||=\max_{\text{eigenvalues } \lambda \text{ of } A}|\lambda|$.

For any square matrix $A$ the eigenvalues of $AA^t$ are all nonnegative definite, so $||A||=\sqrt{||AA^t||}$ is the square root of the largest eigenvalue of $AA^t$.

Using this theorem we can compute the norm of above matrix.  $AA^t=\begin{pmatrix}5&2\\2&1\end{pmatrix}$.  The largest eigenvalue of this matrix is $3+2\sqrt{2}$ and hence $||A||=\sqrt{3+2\sqrt{2}}$, which is below $\sqrt{6}$.