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AXLER’S LINEAR ALGEBRA DONE RIGHT, EXERCISE 1C-24

Posted on July 23, 2020 | Leave a comment

Problem
A function is called even if


   


for all . A function is called odd if


   


for all . Let denote the set of real-valued even functions on and let denote the
set of real-valued odd functions on . Show that:


   



Proof. To show that is a direct sum, it is sufficient to prove that . Let .
Hence, for all , which only happens for . Hence, is a direct sum.

Let . The equation means that can be written as the sum of an odd function and
an even function . Or in other words, for all :


(1)  



and also:


(2)  


By the definition of the odd and even functions, (2) can be written as:


(3)  


The system of linear equations (1) and (3) has a unique solution and . Hence,
can always be uniquely written as the sum of an odd function and an even
function.





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Posted in Axler's Linear Algebra Done Right, Linear Algebra, Mathematics


PINTER’S BOOK OF ABSTRACT ALGEBRA CHAPTER 11, EXERCISE D5

Posted on July 23, 2020 | Leave a comment


Problem
Let be a group and let . Prove the following:
Let , and suppose has a th root, say . Then iff and are relatively prime.

Proof. Consider the cyclic subgroup of . For and , we have , , and . However,
but consists of all even numbers of . Hence . A contradiction.







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Posted in Abstract Algebra, Mathematics, Pinter's Book of Abstract Algebra


PROTECTED: PROGRAMMER BERMAIN TRUF

Posted on April 22, 2016 | Enter your password to view comments.

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Posted in Truf


DP BITMASK

Posted on January 26, 2014 | Leave a comment

Untuk menjelaskan konsep DP Bitmask, saya gunakan salah satu contoh soal di SPOJ
yang saya rasa adalah contoh straight-forward yang bisa langsung dicoba
implementasinya oleh pembaca.
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Posted in Programming Competition


PROBLEM DEVELOPMENT PROCESS

Posted on January 25, 2014 | Leave a comment

Artikel ini dulu saya tulis sebagai guideline dalam persiapan soal di TOKI.
Tetapi saya rasa artikel ini juga akan berguna bagi rekan-rekan yang ingin
mempersiapkan kompetisi-kompetisi pemrograman. Semoga bisa sedikit banyak
membantu meningkatkan kualitas kompetisi pemrograman di Indonesia.

Tahap-tahap pemrosesan sebuah soal yang sudah cukup ideal menurut saya dan biasa
saya terapkan, saya jelaskan sebagai berikut.
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Posted in Programming Competition


OSK – PERSIAPAN

Posted on January 25, 2014 | Leave a comment

Note: Catatan ini ditulis seminggu sebelum OSK 2013

Banyak di antara kita yang karena satu-dua hal baru bisa mulai mempersiapkan
diri untuk mengikuti OSK minggu depan. Tentu saja persiapan yang baik seharusnya
dari jauh-jauh hari. Dulu saya mempersiapkan diri selama kurang lebih satu
tahun, dan menghabiskan sangat banyak waktu. Tetapi, kalau memang tidak
memungkinkan, bukan berarti kita tidak bisa melakukan ‘perlawanan’. Karena ada
beberapa yang minta advice, saya coba rangkum di sini. Semoga membantu.
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Posted in Programming Competition


SENI MEMBUAT BUG, UNTUK KOMPETISI PEMROGRAMAN

Posted on January 25, 2014 | Leave a comment

Peringatan: Apabila Anda tidak mengerti bug itu apa, silakan cari tahu dulu
sebelum membaca lebih lanjut. Kesalahpahaman isi artikel ini akibat mengabaikan
peringatan ini bukan kesalahan penulis.

Pada note ini penulis paparkan tips-tips rahasia bagaimana membuat program penuh
bug berdasarkan pengalaman penulis maupun pengamatan penulis terhadap
orang-orang di sekitar penulis.
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Posted in Programming Competition


DEBUGGING PADA KONTES PEMROGRAMAN

Posted on January 25, 2014 | Leave a comment

Debugging dan implementasi adalah dua hal yang tidak bisa dipisahkan. Seberapa
baik kita bisa melakukan debugging sangat (meskipun bukan seluruhnya) tergantung
dari seberapa baik kita melakukan implementasi. Jadi, tips-tips debugging akan
dimulai dari bagaimana melakukan implementasi agar program kita bisa di-debug
dengan baik. Berikut ini beberapa pegangan saya waktu implementasi (towards
debuggable code) ketika kontes. Tentu saya tidak berharap semua orang setuju
atau cocok dengan tips-tips ini:
Continue reading →

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Posted in Programming Competition

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   * Axler’s Linear Algebra Done Right, Exercise 1C-24
   * Pinter’s Book of Abstract Algebra Chapter 11, Exercise D5
   * Protected: Programmer Bermain Truf
   * DP Bitmask
   * Problem Development Process


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