Miscellaneous Articles for the support of statisticians. It includes different tools and software that help a statistician and a mathematician to work quickly and efficiently.
This post is about some basic introduction to matrix.
Matrices are everywhere. If you have used a spreadsheet program such as MS Excel, or Lotus, written a table (such as in Ms-Word), or even have used mathematical or statistical software such as Mathematica, Matlab, Minitab, SAS, SPSS, Eviews, etc., you have used a matrix. Let us start with the Introduction to matrix.
Introduction to Matrix
Matrices make the presentation of numbers clearer and make calculations easier to program. For example, the matrix is given below about the sale of tires in a particular store given by quarter and make of tires.
Firestone
Q1
Q2
Q3
Q4
Tirestone
21
20
3
2
Michigan
5
11
15
24
Copper
6
14
7
28
It is called a matrix, as information is stored in a particular order and different computations can also be performed. For example, if you want to know how many Michigan tires were sold in Quarter 3, you can go along the row ‘Michigan’ and column ‘Q3’ and find that it is 15.
Similarly, the total number of sales of ‘Michigan’ tiers can also be found by adding all the elements from Q1 to Q4 in the Michigan row. It sums to 55. So, a matrix is a rectangular array of elements. The elements of a matrix can be symbolic expressions or numbers. Matrix $[A]$ is denoted by;
Row $i$ of the matrix $[A]$ has $n$ elements and is $[a_{i1}, a_{i2}, cdots, a_{1n}] and column of $[A]$ has $m$ elements and is $begin{bmatrix}a_{1j}\ a_{2j} \ vdots\ a_{mj}end{bmatrix}$.
The size (order) of any matrix is defined by the number of rows and columns in the matrix. If a matrix $[A]$ has $m$ rows and $n$ columns, the size of the matrix is denoted by $(mtimes n)$. The matrix $[A]$ can also be denoted by $[A]_{mtimes n}$ to show that $[A]$ is a matrix that has $m$ rows and $n$ columns in it.
Each entry in the matrix is called the element or entry of the matrix and is denoted by $a_{ij}$, where $i$ represents the row number and $j$ is the column number of the matrix element.
The above-arranged information about sales and types of tires can be denoted by the matrix $[A]$, that is, the matrix has 3 rows and 4 columns. So, the order (size) of the matrix is 3 x 4. Note that element $a_{23}$ indicates the sales of tires in ‘Michigan’ in quarter 3 (Q3). That is all about Introduction to Matrix.
The basics of algebra include numbers, variables, constants, expressions, equations, linear equations, and quadratic equations. Further, it involves the basic arithmetic operations of addition, subtraction, multiplication, and division within the algebraic expressions.
Introduction to Algebra
We work with numbers in arithmetic, while in algebra we use numbers as well as Alphabets such as $A, B, C, a, b$, and $c$ for any numerical values we choose. We can say that algebra is an extension of arithmetic. For example, the arithmetic sum of two numbers $5+3=8$ means that the sum of numbers 5 and 3 is 8. In algebra, two numbers can be summed by the expression $x+y=z$ which is the general form that can be used to add any two numbers. For example, if $x=5$ and $y=3$ then $x+y$ will be equivalent to the left-hand side ($5+3$) and the summation of these numbers will be equivalent to the right-hand side $z$ which is 8.
In algebra all arithmetic operators such as $+, -, \times, =$ and $\div$, etc., can be used used.
For example, $x-y=z$ means that the difference between two numbers is equal to the number represented by the letter z. In algebra, many other notations used are the same as in arithmetic. For example,
$c=a\times b$ means that the product of two numbers represented by $a$ and $b$ is equal to the number $c$.
$x \times x \times x$ can be written as $a^4$.
From the above discussion, note that letters of the alphabet represent variables, and arithmetic operators (+, -, etc) represent the mathematical operations on a variable. The combination of numbers and letters of the alphabet is called an algebraic expression. For example, $8x + 7y$, $x+y$, and $7x^2+2xy-5y^2$ etc. are examples of expression.
Some important points to remember:
Algebra is like a toolbox for solving mathematics problems with unknowns. Instead of using specific numbers, we use letters like $x$, $y$, and $z$ to represent unknown values. These letters are called variables.
A variable is a quantity (usually denoted by letters of the alphabet) in algebraic expressions and equations, that changes from place to place, person, to person, and/or time to time. The variable can have any one of a range of possible values.
A factor that multiplies with a variable. For example, in $2x^3+3x=0$, $x$ is a variable, 2 is the coefficient of $x^3$, and 3 is the coefficient of $x$.
We learning algebra the following concepts are very important to understand the concepts used in algebra:
Variables are the building blocks, representing unknown numbers.
Expressions are combinations of variables, numbers, and mathematical operations (such as +, -, *, /) that do not necessarily have an equal sign (=).
Equations are statements (or expressions) with an equal sign that shows two expressions are equivalent. The equations are solved to find the value of the variable.
Inequalities are statements (or expressions) having “greater than” (>), “less than” (<), or “not equal to” (≠) symbols for making comparisons between expressions.
Real Life Applications of Algebra
Finances and Budgeting: Algebra helps you create formulas to track income, expenses, savings goals, and loan payments. You can set up equations to see how much extra money you’d have if you cut certain expenses or how long it’ll take to save for a down payment on a house.
Mixing and Ratios: Whether you’re baking a cake or mixing paint colors, algebra helps you determine the correct ratios of ingredients to achieve the desired outcome. You can set up proportions to find out how much water to add to a paint concentrate or how much flour you need to double a recipe.
Motion and Physics: From calculating travel time based on speed and distance to understanding the trajectory of a thrown ball, algebra forms the foundation for many physics concepts. You can use formulas to figure out how long it’ll take to drive somewhere at a certain speed or the angle needed to throw a basketball into the hoop.
DIY Projects and Home Improvement: From measuring lumber for a bookshelf to calculating the amount of paint needed for a room, algebra helps with planning and executing home improvement tasks. You can use formulas to find the area of a wall to determine how much paint to buy or calculate the volume of wood needed for a project.
Scientific Research and Data Analysis: Algebra is the backbone of many scientific formulas and equations used in research. It helps analyze data, identify trends, and make predictions.
Estimation (Approximating a Precise Value) is very useful especially when someone wishes to know whether he/ she has arrived at a logical solution to a problem under study. It is useful to learn about how to estimate the total sum of a bill to avoid immediate overpayments. For example, one can estimate the total amount of shop (supermarket) receipts. The estimate of these receipts can be done by rounding the amount of each item to the nearest half and keeping a running total mentally from the first item to the last one.
Estimation of a Utility Bill
Suppose the following is a shop receipt, with the estimated amount and running total. Consider, the estimation, approximating a precise value for a utility bill. Shop Item, Actual Amount, Estimated Amount, Running Total.
Shop Item
Actual Amount
Estimated Amount
Running Total
Item 1
4.50
4.50
4.50
Item 2
3.50
3.50
8
Item 3
1.3
1.5
9.5
Item 4
0.60
0.5
10
Item 5
2.95
3
13
Item 6
2.85
3
16
Item 7
1.60
1.50
17.5
Item 8
2.75
3
20.5
Item 9
2.4
2.5
23
Total
22.45
23
From the above example, it can be observed that estimation is a process of finding an estimate of a value. It saves time and results in the nearest possible exact value. An estimate can be overestimated (when the estimate exceeds the actual value) and underestimated (when the estimate falls short of the actual value).
In some cases, an estimate can be performed to round all of the numbers that you are working to the nearest 10 (or 100 or 1000) and then do the necessary calculations. In everyday life, the estimation can be used before you solve a problem in an easier and faster way. It helps you to determine whether your answer is reasonable. Estimation is also useful when you need an approximate amount instead of a precise value.
