0
votes
0answers
39 views

Invariant subspaces

Given the subspace $U = \operatorname{span}\left( \left\{ \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix},\begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} \right\} \right)$ and the following linear map $t : ...
1
vote
0answers
29 views

A set is partitioned into $k$ non-empty sets, and the difference between elements of each subset is given, reconstruct each subset

We have a set $A = \{1, 2, 3, \dots, n\}$. This set is partitioned into $k$ non-empty subsets: $$A_1 = \{a_1, a_2, \dots, a_{m_1}\}$$ $$A_2 = \{a_{m_1 + 1}, a_{{m_1} + 2}, \dots, a_{m_2}\}$$ $$.$$ ...
0
votes
1answer
77 views

Pattern for Recursive Construction

Suppose I have this recursive definition of binary strings. Let $K$ be set of binary strings. The empty string $""$ and $1$ are in $K$. If $k$ is a string in $K$, then so is $0k$, $k0$. And if $k$ is ...
1
vote
0answers
72 views

How to show that the Harris corner detector is rotation invariant

I understand what it means for the Harris corner detector to be rotation invariant. But what steps do you need to take to show that it is rotation invariant?
0
votes
3answers
60 views

Prove that an algorithm cannot reach a given goal

We are given an algorithm that, in each step takes a set $\left\{a, b, c\right\}$ It takes any two variables $a, b$ at random and changes them to $0.6 + 0.8b$ and $0.8a - 0.6b$. The initial value of ...
2
votes
2answers
26 views

Prove that n is divisible by 4 in a cylic sum with variables which have only two possible values

It is known that $a_1, a_2, a_3, ... , a_n \in \left\{-1, 1 \right\}$ and $S = a_1a_2a_3a_4 + a_2a_3a_4a_5 + ... + a_na_1a_2a_3 = 0$ Prove that $n \equiv 0\space(mod\space 4)$ I know this problem ...
3
votes
3answers
84 views

Numbers in a sequence

The number sequence $1, 9, 8, 2...$ satisfies the following rule: each element of the sequence starting from the fifth, is equal to the last digit of the sum of the previous four members. Will we ...
1
vote
1answer
58 views

Prove a transformation is a variational symmetry for J

The following problem is from The Calculus of Variations by B.von Brunt (page 215, Exercise 9.2.1) Let $$ J(y)=\int_a^b xy'^2\mathrm{d}x. $$ Show that the transformation $$ X=x+\epsilon2x\ ...
-1
votes
1answer
65 views

Variance Formula

for the variance formula $\text{var}(X) = E[X^2]-(E[X])^2$ how are you suppose to work out $E[X^2]$ given the interval $[0,1]$ and $E[x]= 1/2$