Matematik merupakan perkara penting yang perlu dipelajari dalam kehidupan

manusia. Sejak dari kecil lagi ada kanak-kanak yang sudah belajar mengira dan

melukis sesuatu obejek. Adakah anda sedar bahawa semua fenomena ini adalah

merupakan persediaan yang penting dalam mempelajari algebra? Keupayaan dalam

algebra akan membantu manusia berjaya dalam bidang yang kita ingin ceburi.

Pelbagai bidang di dalam dunia ini mempunyai kaitan dengan pembelajaran algebra.

Mungkin antara kita yang tidak tahu dan tidak sedar akan kewujudan ‘linear algebra’

dalam kehidupan kita.

Melalui tugasan ini, kami akan merungkaikan aplikasi linear algebra dalam

kehidupan manusia. Antaranya ialah:

KEWANGAN

Adakah anda sedar yang anda menggunakan linear algebra dalam

merancang kewangan seharian anda? Sebagai contoh, linear algebra banyak

diaplikasikan semasa anda meminjam duit dan membuka akaun untuk membayar

bunga. Fenomena ini sering berlaku dalam kalangan rakyat Malaysia dalam proses

membeli kereta atau rumah. Mereka akan membuat pinjaman bank untuk membeli

kereta atau rumah terlebih dahulu. Kemudian, mereka akan membayar hutang

tersebut secara bulanan.

Untuk merancang kewangan anda dalam pembayaran kereta. Kita boleh

menggunakan formula ini:  

Pmt = L x [r x ( (1+r)^n)]/ [( ((1+r)^n))-1]

L ialah jumlah pinjaman

r ialah bunga setiap bulan

n ialah tempoh bulan pembayaran.

Sekiranya anda membeli kereta yang berharga $30,000, dengan APR of 9%,

tempoh pembayaran 60 bulan.

Bayaran bulanan anda ialah $622.75 untuk 59 bulan, dan pembayaran terakhir anda

ialah $622.79 untuk bulan terakhir.

Jumlah yang anda akan bayar ialah $37,365.04.

Jumlah bunga ialah $7,365.04.

Sekiranya anda menyelesaikan persamaan ini untuk n, dan menukarkan Pmt + A

untuk Pmt di mana A ialah additional principle payment. Anda dapat membayar

semula pinjaman anda lebih awal dan jimat dalam pembayaran bunga.

Di bawah merupakan contoh lain bagi menunjukkan aplikasi linear algebra.

Contoh di bawah menerangkan berapa jumlah barang yang kita dapat beli dengan

jumlah wang yang kita ada.

Anda pergi ke kedai serbaneka dan mempunyai $10.00 dan anda ingin membeli

coklat yang berharga $2.00. Berapakah jumlah coklat yang boleh anda beli dengan

wang sebanyak $10.00?

Persamaannya ialah 2x = 10 di mana x ialah jumlah coklat yang boleh dibeli dengan

wang sebanyak $10.00.

Kebanyakkan orang tidak sedar tentang pengiraan yang dilakukan mereka adalah

algebra.

SOSIOLOGI

Sociologists interested in various kinds of communications in a group of individuals

often use graphs to represent and analyze relations inside the group. For

terminology and some results about graph theory that we will use here, check the

application of linear algebra to graph theory. The idea is to associate a vertex to

each individual in the group, and if individual A influences or dominates individual B,

we draw a directed edge from A to B. Note that the obtained graph can have at most

one directed edge between two distinct vertices.

Contoh

Bayangkan satu kumpulan mempunyai lapan individu, iaitu I1,…, I8. Digraf di bawah

mewakili hubungan dominan antara setiap individu dalam kumpulan tersebut:

The adjacency matrix of this graph is:

The row with most 1’s in the above matrix corresponds to the most influential individual in the group; in our case it is I6.  In the above graph, walks of length 1 (one edge) correspond to direct influence in the group, whereas walks of greater length correspond to indirect influence.  For instance I3 directly influences I5 and I5 directly influences I4, therefore I3 indirectly influences I4.

Now, squaring M gives

One can see that individual I8 has 2-stage influence on half of the group, although he has only one direct influence on I6.

FOTOGRAFI

Different digital cameras have different spectral sensitivities and opto-electronic conversion functions (OECF), and therefore produce different data about the same scene. Correct interpretation of the data requires that it be presented in some sort of standard form. The obvious choice for a standard spectral space is a spectral space spanned by a set of color matching functions, a color space.3x3 matrices are used for transformations from camera spectral spaces to a standard color space based on the ITU-R BT.709 red, green, and blue (RGB) primaries.

3x3 matrix transformations are used most because such transformations are most appropriate when the relationship between the scene radiance and the radiance incident on the sensor is variable, different illumination sources are used, and the colorants found in the scene are unknown or highly variable.

The first step in determining the desired transformations is to linearize the data obtained from the camera with respect to the source or scene radiance.

For this experiment, the camera data was linearized by measuring the camera and focal plane OECF’s of the

camera for a variety of scene dynamic ranges and mean reflectances, and constructing a flare model which predicts the camera flare based on focal plane image statistics.

Accurate OECF measurements for camera data linearization are extremely important, particularly with the LS methods, since the regression tries to transform the chart image data to aim linear values.

The numerical results of the experiments conducted are presented in the following table:

(WPPLS): white point preserving LS regressions.(WTWPPLS):  weighted white point preserving LS regressions.

KRIPTOGRAFI

Kriptografi ialah salah satu contoh aplikasi linear algebra dalam satu bidang

Matematik. Ia meruapakan satu kajian tentang penyulitan (encryption) dan

penyahsulitan (decryption) mesej. Mesej tanpa penyulitan dipanggil plaintext,

manakal mesej yang tersulit dipanggil chiphertext. Algoritma untuk proses penyulitan

dan penyahsulitan dipanggil chipher. Salah satu jenis chipher yag melibatkan

aplikasi linear algebra ialah Hill Chipher. Untuk pengetahuan semua, kriptografi ini

amat penting dan digunakan untuk keselamatan maklumat sulit pengguna seperti

akaun email, kad kredit dan kad pengenalan daripada disalahgunakan oleh pihak

yang tidak bertanggungjawab. Tambahan pula, kriptografi ini juga telah digunakan

dalam bidang ketentaraan dari segi komunikasi dan isyarat.

Contoh:

Plain text: Time is running out..

Mesej ini akan ditukarkan kepada mesej baru dengan menggunakan Mod 29.

Selepas menggunakan Mod 29, anda akan dapat mesej baru seperti di bawah:

Table mod 29

Chipher text: N , , . C O J J . , H , X I C V H V E O T

Bagi menyahkan chipher text, anda memerlukan ‘Multiplicative Inverse Mod 29’.

Table ‘Multiplicative Inverse Mod 29’.

important

Algebra is hard enough to understand, and now they want you to use and understand linear algebra. What is it good for and why should you learn it? How will you be able to use it in the future? These are good questions and this should provide some of the answers.

Linear algebra is typically used when making a graph, and compares the changes in a variable compared to a set time or distance or other constant. Most linear equations will for a straight line graph. As an example, this can be used to determine how far a vehicle will travel at a fixed rate of speed at various time intervals. After you plot out the graph, you can determine the unknown variable (distance) by plotting it on the graph. This can be used for a multitude of different functions and is a handy tool for lots of different real life functions.

Understanding linear equations is a necessary stepping stone to understanding more complex algebra and calculus equations and graphing capabilities. And by putting the equation into a graphical form, it can be more easily understood and interpolated what the unknown value is

at a quick glance. This is something that can be used in baking, real estate, construction and just about every job in existence.

Like anything new, it can be a little complex when you first start learning it. However, once you grasp the concepts, it literally can be an almost intuitive tool that you wonder how you managed to function without it.

Before you can start using linear algebra you need to have a basic understanding of algebra and algebraic equations. This is also one of the primary building blocks for advanced calculus. It is important to understand how to build the simple graphs of linear equations before you can get into the more complex 3-D modeling.

If you do have trouble understanding it when you first get started, don't panic. Ask your instructor for help. If your text book doesn't make it clear, you may need to get a supplemental text to help. You can find supplemental text books in your school library or online.

Even though linear algebra sounds complicated, it truly is one of the most important building blocks in advanced math and is definitely one you need to master. So take the time and effort to learn it and work the additional problems so you can completely understand it.