pps matrikulkasi aljabar 2015
DESCRIPTION
Rencana perkuliahan aljabar linear s2TRANSCRIPT
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Universitas Konservasi
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Lambang Almamater
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Aljabar Elementer
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Aljabar
• Salah satu cabang utama matematika, selain geometri, analisis, topologi, teori bilangan, dan kombinatorik
• Cabang matematika yang mengkaji aturan/hukum-hukum operasi dan relasi, serta konstruksi dan konsep yang muncul dari operasai dan relasi, mencakup term, polinomial, persamaan, struktur aljabarnya.
• a generalization of arithmetic in which letters representing numbers are combined according to the rules of arithmetics
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• the branch of mathematics that deals with general statements of relations, utilizing letters and other symbols to represent
specific sets of numbers, values, vectors, etc in the description of such relations.• any of various systems or branches of mathematics or logic concerned with the properties and relationships of abstract entities (as complex numbers, matrices, sets, vectors, groups, rings, or fields) manipulated in symbolic form under operations often analogous to those of arithmetic
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Algebraic thinking• Algebraic thinking is about generalising arithmetic operations and operating on unknown quantities. It involves recognising and analysing patterns and developing generalisations about these patterns. In algebra, symbols can be used to represent generalisations.
• For example, a + 0 = a is a symbolic representation for the idea that when zero is added to any number it stays the same. Studying and representing relationships is also an important part of algebra.
• "The language of arithmetic focuses on answers, while the language of algebra focuses on relationships."1
MacGregor, M & Stacey, K. (1999) “A flying start to algebra. Teaching Children Mathematics, 6/2, 78-86. Retrieved 17 May 2005
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Aljabar Elementer
• Aljabar yg mengenalkan konsep variabel yg mewakili bilangan. Pernyataan-pernyataan yang berdasarkan pada variabel-variabel itu dimanipulasi dengan menggunakan aturan-aturan operasi yang menggunakan bilangan, yang utama yaitu penambahan, pengurangan, perkalian dan pembagian. Ini dapat dilakukan untu berbagai pertimbangan, termasuk penyelesaian persamaan.
• Biasanya menjadi bagian dalam kurikulum sekolah.
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Elementary AlgebraThree categories of questions
• operations with integers and rational numbers, includes computation with integers and negative rationals, the use of absolute values, and ordering.
• operations with algebraic expressions (skills in evaluating simple formulas and expressions, and in adding and subtracting monomials and polynomials).
Both of the preceeding categories include questions about multiplying and dividing monomials and polynomials, evaluating positive rational roots and exponents, simplifying algebraic fractions, and factoring.
• skill in solving equations, inequalities, and word problems (solving systems of linear equations, quadratic equations by factoring, verbal problems presented in algebraic context, geometric reasoning, the translation of written phrases into algebraic expressions, and graphing).
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Aljabar dan Aljabar Elementer
• Algebra lebih luas dari pada aljabar elementer. • Aljabar mengkaji apa yang terjadi ketika aturan yang bberbeda dari suatu operasi digunakan dan ketika operasi direncanakan untuk digunakan untuk sesuatu selain bilangan
• Penjumlahan dan pengurangan dan digeneralisasikan dan definisi yang mirip untuk membangun struktur-struktur aljabar seperti grup, ring, field, yang mempelajari bidang matematika yang disebut Aljabar Abstrak.
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hasil kegiatan manusia yang terorganisir,
suatu konstruksi sosial,
suatu hasil budaya,
pengetahuan yang mungkin salah
HAKIKAT MATEMATIKA
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• abstrak, teoritis, non empiris,• umum, objektif, formal, rational
• konsep epistemologis self evidensi, deduktif aksiomatik, • artificial,
• kebenarannya tidak mutlak .
KARAKTERISTIK MATEMATIKA
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undefined term
konsep,
aksioma,
teorema
KOMPONEN MATEMATIKA
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Sistem Matematika
Teorema
(dibuktikan)
deduktifdeduktif
Defined term
(didefinisikan)
Aksioma
(ditetapkan)
kesepakatankesepakatan
unsur primitive/
undefined term
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Cara Memahami Definisi1. Menganalisis rumusan definisi atas:
- Latar belakang, konteks, ketentuan dasar,- Subjek atau objek pembicaraan- Istilah - Ungkapan (yang didefinisikan)- Atribut (biasanya merupakan suatu gugus kalimat)- Simbol yang digunakan
2. Mengingat kembali - istilah/konsep yang ada dalam definisi- teorema-teorema yang terlibat dalam definisi
3. Memberi contoh objek yang - memenuhi rumusan definisi- tidak memenuhi rumusan definisi (noncontoh)
4. Membuktikan objek yang sudah diketahui memenuhi rumusan definisi
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Contoh
Definisi Misalkan a suatu bilangan real. Harga mutlak dari a ditulis |a|didefinikan oleh |a| = a, jika a≥0 -a, jika a<0
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Menganalisis definisi
- LB : R, a bilangan real- Subjek : elemen-elemen R- Istilah : harga mutlak- Ungkapan : X adalah harga mutlak a- Simbol : X = |a|- Atribut : X = |a| = a, jika a≥0 -a, jika a<0
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Barisan Bilangan
• Definisi 3.1.1 A sequence of real number ( or sequence in R) is a function on the set N of natural number whose range is contained in the set R of real number.
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Menganalisis definisi
- LB : R, N- Subjek : f (fungsi)- Istilah : barisan- Ungkapan : f adalah barisan bilangan real- Atribut : f: N → R- Simbol : X atau (xn)
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Konsep yang terlibat
• Fungsi• Limit fungsi• ...
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Contoh dan noncontoh
• 2, 4, 6, 8,.... (barisan)• ½, ¼, 1/8, ... (barisan• f: N→ {1,-1} (barisan)• 2,4,8,10, 12. (bukan barisan)• f: R→ R (bukan barisan)• f: Q→ R (bukan barisan)
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Himpunan Rentangan
Misalkan vektor-vektor v1, v2, v3, ...vn dalam V.
S = { v1, v2, v3, v4, ...vk }.
W disebut himpunan rentang dari S (S merentang W), jika
W= himpunan semua kombinasi linear dari v1, v2, v3, ...vk
.
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Menganalisis definisi
- LB : V ruang vektor- Subjek : S himpunan bagian V- Ungkapan : S merentang W- Atribut : W = himpunan semua kombinasi linear
dari anggota-anggota S
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Mengingat kembali istilah/konsep yang ada dalam definisi
Ruang vektor
Kombinasi linear
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Cara Memahami Teorema
1. Menganalisis teorema atas- Latar belakang, konteks, ketentuan dasar,- Hipotesis, premis, antiseden- Konklusi, kesimpulan, konsekuen
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Cara Memahami Teorema(lanjutan)
2. Mengingat kembali - istilah/konsep yang ada dalam teorema,- teorema-teorema yang terlibat dalam teorema
3. Menjajagi corrolies (teorema akibat) dan merumuskannya
4. Menganalisis bukti dengan menunjukkan- langkah utama,- langkah rinci dari setiap langkah utama,- alasan/justifikasi yg dipakai,- teorema yang digunakan
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Contoh
Teorema|a|=0 jika dan hanya jika a=0
Bukti:Jika a=0, maka |a|=a=0Jika a≠0, maka -a ≠ 0, sehingga |a|≠0. Jadi, jika |a|=0, maka a=0
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Menganalisis Teorema
- Latar belakang, konteks, ketentuan dasar: R- Hipotesis, premis, antiseden: Bagian pertama: |a|=0 Bagian kedua : a=0- Konklusi, kesimpulan, konsekuen:
Bagian pertama a=0 Bagian kedua |a|
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Konsep yg terlibat
Harga mutlak
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Menganalisis Bukti
• Langkah utama 1. membuktikan bahwa jika a=0, maka |a|=0 2. membuktikan bahwa jika |a|=0 maka a=0
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Memberi Justifikasi
Teorema|a|=0 jika dan hanya jika a=0
Bukti:Jika a=0, maka |a|=a=0 ......definisi harga mutlakJika a≠0, maka -a ≠ 0..........sifat bilangan Sehingga |a|≠0............. definisi harga mutlak jadi, jika |a|=0, maka a=0...........