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## Math 487, midterm, solutions

\(\def\Real{\mathbb{R}} \def\image{\mathtt{Im}}\)

- [10] Find the dimensions of image and kernel for the operator \(A:\Real^5\to\Real^4\) given by the matrix

$$

A=\left(\begin{array}{ccccc}

0&1&2&3&4\\

1&2&3&4&5\\

2&3&4&5&6\\

3&4&5&6&7

\end{array}\right).

$$*If one chooses the basis*

$$

e’_1=e_1, e’_2=e_2, e’_3=e_3-2e_2+e_1,e’_4=e_4-2e_3+e_2,e’_5=e_5-2e_4+e_3,

$$

one sees immediately, that

## Fact of the day: Brownian centroids

\(\def\Real{\mathbb{R}}\def\Z{\mathcal{Z}}\def\sg{\mathfrak{S}}\def\B{\mathbf{B}}\def\xx{\mathbf{x}}\def\ex{\mathbb{E}}

\)

Consider \(n\) points in Euclidean space, \(\xx=\{x_1,\ldots, x_n\}, x_k\in \Real^d, n\leq d+1\).

Generically, there is a unique sphere in the affine space spanned by those points, containing all of them. This centroid (which we will denote as \(o(x_1,\ldots,x_n)\)) …

## Math 487, 9.17

Underlying text: DiAngelo’s book, chapter 1.

- Fields, – definition, examples: \(\mathbb{Q, R, C, Z}_p\).
- General linear spaces (matrices, functions with values in a field).
- Linear operators.

#### Exercises:

- Find the image of the linear operators \(A,B:U\to U\) on \(U\), the

## ECE 515, 9.13

Lecture notes, 3.1-3.5.

- Solutions of non-homogenous linear systems;
- Exponentials of diagonal and diagonalizable operators and matrices;
- Lagrange approximations and reduction of matrix functions to polynomials;
- Non-autonomous systems and Picard approximations.

## ECE 515, 9.11

Lecture notes, 3.1-3.5.

- Cayley-Hamilton Theorem;
- Matrix exponentials;
- Solutions to linear systems differential equations.

## Math 487, Homework 1

\(\def\Real{\mathbb{R}}\def\ee{\mathbf{e}}\def\ff{\mathbf{f}}\newcommand\ket[1]{\vert{#1}\rangle}\newcommand\bra[1]{\langle{#1}\vert}\newcommand\bk[2]{\langle #1\vert#2\rangle}\def\lin{\mathtt{Lin}} \)

**(15)**Find all matrices

$$

X=\left(

\begin{array}{cc}

x&y\\

z&w

\end{array}

\right)

$$

such that \(XA=BX\), where

$$

A=\left(

\begin{array}{cc}

1&2\\

-1&0

\end{array}

\right), \mathrm{\ and\ }

B=\left(

\begin{array}{cc}

0&1\\

3&0

\end{array}

\right).

$$**(15)**Same for

$$

## ECE 515, 9.6

Following lecture notes, 2.6, 2.7:

- Eigenvalues and eigenvectors;
- Operators with distinct eigenvalues; diagonalization;
- Failure to diagonalize leads to considering chains of subspaces \(V^{(k)}_\lambda:={\mathtt{Ker}} (A_\lambda)^k, A_\lambda:=A-\lambda E\), which lead to
- Jordan normal form.

## Math 487, 9.5

Continuing to use Olver’s book, chapter 1, but also relying on Prasolov’s notes on linear algebra.

- Determinant (as a function from \({\mathtt{Mat}}(n\times n;\mathbf{k})\): existence and properties.
- Matrix is singular iff its determinant vanishes.

## Fact of the day: Condorset domains, tiling permutations and contractibility

\(\def\Real{\mathbb{R}}\def\Z{\mathcal{Z}}\def\sg{\mathfrak{S}}\def\B{\mathbf{B}}

\)

Consider a collection of vectors \(e_1,\ldots,e_n\) in the upper half-plane, such that \(e_k=(x_k,1)\) and \(x_1>x_2\gt \ldots \gt x_n\). Minkowski sum of the segments \(s_k:=[0,e_k]\) is a zonotope \(\Z\). *Rhombus* in this context are the Minkowski sums \(\Z(k,l)=s_k\oplus s_l, …