Learning Linear Algebra

The study of linear sets of equations and the transformations they undergo together is called Linear Algebra. While Learning Linear Algebra we come across many topics like, Systems of Equations, Matrices, Determinants, Euclidean n-space, Vector Spaces and Eigen Values & Eigen Vectors. These topics can be learnt with ease using many courses offered as online course Linear Algebra.
Math Linear Algebra
In this article we shall learn in brief about system of equations and matrices. The general linear equation is given by a1x1+ a2x2+ a3x3+…..+ anxn = b; here there are n unknowns and the known numbers are x1, x2, x3…xn . The solution set of which would be given by the set of numbers s1, s2, s3….., sn such that if we equate x1= s1, x2=s2, x3=s3….., xn=sn then the equation is satisfied.

 This means that when the solution set values are substituted on the left hand side of the equation that would be equal to the value b on the right hand side.  Solving system of equations involves different methods in which use of matrices is one of them. Some of the methods are used in solving the systems of linear equations are Gaussian Elimination and Gauss-Jordan Elimination. A series of steps in this method help in solving the given equations.

Let us now take a quick look at Matrices, a matrix is a rectangular array of numbers in the form of rows and columns and each of these elements is called an ‘entry’. The size of any given matrix is denoted with n rows and m columns using nxm. For instance, 3x4 shows it is a matrix consisting of 3 rows and 4 columns.
The matrix that has only one column is called a column matrix (vector) and the one which has only one row is called a row matrix. A matrix is denoted using an upper case letter and the entries using lower case letters. The entry in the ith row and jth column is denoted as ‘aij’ when the matrix considered is ‘A’. A square matrix is the one which has equal number of rows and columns and is denotes as nxn.
In a square matrix the entries a11, a22…ann which form the numbers in a diagonal is called the main diagonal of the matrix. This article gives a very brief outline of Matrices. To learn and understand the various topics under linear algebra with ease and to get Linear Algebra homework help there are many online courses that are offered to one and all.

Sequence numbers

Informally, if we want to define sequence (seq) it will be something as an arrangement of events, elements, terms etc. It is a manner or discipline to keep the things in order. It may be a set of members. The length of the same is determined by the number of ordered elements. It is the arrangement of similar objects. The same elements can appear many times at different positions in the same arrangement.

Sequence definition:
A function f(x) which has domain and range, where x may be set of the natural numbers is called as its definition. The seq are of the following types.
( 1) Finite
(2) Infinite
Finite seq:- In finite form the number of elements is countable. For example
A= ( 1,3,5,7,……………111)
B =(2,4,6,8………………112)
Seq of any finite length ‘n is termed as an n-tuple and Finite seqs might also include empty form ( ) which will have no elements.
Infinite seq:- In the Infinite form the number of elements is not countable. For example
A =(……….. -3,-2,-1,0,1,2,3……………….)
Infinite seq is infinite in both directions. It has neither a first nor a final element is called a bi-infinite or two-way infinite. For example a function from all the integers included into a set, such that the seq of all even integers ( …-8,-6 -4, -2, 0, 2, 4, 6, 8,10,12… ), is found to be bi-infinite.
There are many important integer sequential forms and these are as follows.
(A) The even numbers which can be divided by 2.
(B) The odd numbers which cannot be divided by 2.
(C) The prime numbers that have no divisors except 1 and themselves.
The Fibonacci number Sequences:- It is nothing but in which elements are the sum of the previous two elements. The first two elements are either 0 or 1. This is (0,1,1,2,3,5,8,13,21,34,65,99...).
Formula for the Fibonacci:- It  can be defined using a recursive rule along with two initial elements.
 ,   with   a0 = 0  and  a1 = 1.
Where, 0 and 1 are initial elements of the Fibonacci sequence.

Special seq:- some of the special seq forms are given below.

(1) Arithmetic
(2) Geometric
(3) Square of numbers
(4) Triangular
(5) cube of numbers
(6) Roots of numbers
(7) cubic roots of numbers
(8) A set of vowels.
(9) indexing of the documents
Examples and notation
It is a list of elements with a particular order. These are useful for the study of the functions, spaces, and other mathematical structures by using the properties of convergence . The basis for series is sequences. These are used in differential equations and analysis. These are also used to find the patterns or to solve the puzzles and can be used in the study of prime numbers.

Solving simple equations

Simple equations are equations those have just one variable and which can be solved by using very algebraic operations. At times, you can even solve them mentally. These are also called one step equations because in one step. Since they involve variables and need to perform algebraic operations for solutions, they are also referred as algebra simple equations.

In many cases the one step equations are based on word problems and those are framed as per the statement of the word problem. Let us illustrate a few examples to understand how to solve simple equations.

Let x + 1 = 5. The solution is to find the value of x. We see on the left side a 1 is added to the variable. So to isolate that, we need to undo that addition by the inverse algebraic operation, which is subtraction. Since 1 is added to the variable, in the undoing process we need to subtract 1 on the left side. But since this is an equation doing any operation only on one side is prohibited since the balance of the equation is disturbed. Hence to maintain the balance a 1 must also be subtracted on the right side.

So the method of solution is (x + 1) – 1 = 5 – 1, or, x = 4, which is the solution. The solution can be checked by plugging that value in the original equation. That is, if x = 4, then the left side becomes as, x + 1 = 4 +1 = 5 = right side. Hence the solution is correct.

Let x – 3 = 4. Here the undoing operation is ‘add 3’ on both sides. So, (x – 3) +3 = 4 + 3, or, x = 7 which is the solution.
Next let us the equation 3x = 9. In this case, the variable is multiplied by 3 on the left side. In this case, the undoing operation is ‘divide by 3’ on both sides. So, (3x/3) = (9/3) or, x = 3, which is the solution. Similarly the undoing operation for a division is ‘multiply by the same number’ on both sides. For example, if, (x/2) = 4, then, (x/2)*2 = 4*2 or, x = 8.

Let us see how one step equations are formed from word problems.
‘Ben is 5 years older than Joan and Joan is 3 year old. How old is Ben?’ Let ‘x’ be the age of Ben. Since he is 5 years older that Joan, the equation is x – 5 = 3 and the solution is x – 5 + 5 = 3 + 5 or x = 8. Hence Ben is 8 years old.

Templates for the Diagrams Representing Set Operations

The Venn diagram template can be associated with set theory in mathematics. Sets represent collection of objects which are similar in nature. So, the all real numbers together form a set. Even the integers together can form a set. These are similar numbers which is necessary to form a set. The triple Venn diagram template is used when three sets intersect.

There is a common area when the sets intersect. There are various operations that are to be learnt in the set theory. The operations like union or intersection can be used to solve various problems. The set theory is used to solve the problems in a simpler manner.

They give a clear understanding of the problems. The 3 circle Venn diagram template is used when three sets are intersecting. These templates can be very useful and helpful. They must be chosen carefully.

The Venn diagram templates are used to make these better understandable so that the problems can be better understood and can be solved easily. Even Free Venn diagram templates are available online and can be chosen at will to solve the problems. They can be really helpful. A wide variety of them are available and the right choice must be made. For this to happen one must be thorough with the concepts in set theory and these diagrams.

These diagrams are pictorial representations of the problems in the set theory. Once they are drawn one is able to understand the problem better and can solve it more easily and in a smaller amount of time. So, one must learn how to draw these diagrams.

The intersection of two sets in these diagrams denotes common area between the two sets. It contains the common elements which are present in both the sets. So, the common area in these diagrams is the intersection and denotes the elements which are present in both the sets.

The same can be true if there more than two sets. In case of three sets also this is the case. The common area denotes the elements which are present in all the three sets. The union of sets denotes all the elements present in the sets of which the union is found. There is also the concept of empty set. If there is no common area in the diagram, then it denotes an empty set. It basically means there are no common elements between the sets.

Basics of Simple Interest

The term interest refers to the cost of borrowing money. The interest calculation varies from plan to plan and is also based on the lenders and the time period of lending/depositing the amount. This interest is calculated in different ways such as interest only on principal, interest on principal and interest so far earned/incurred, monthly interest, cumulative interest etc.  The amount for which the interest is calculated can be a loan (amount borrowed for need) or a deposit (amount deposited as savings).

Definition
The interest which is calculated only on the principal amount borrowed or deposited is termed to be Simple Interest that is denoted as SI in short.  This type of interest does not include the interest so far incurred or earned on the principal amount.

When an amount is borrowed, the amount borrowed is called the Principal. The duration which the borrower takes to return the Principal is termed as the time period and is calculated in number of days/months/year.  The next and most important part is the rate of interest which states the interest percentage for the given principal amount.  All the three put together explains what is Simple Interest. The interest percentage is directly proportional to the lender and the time duration to repay the amount.   Also it depends upon whether it is a loan or a deposit.

Formula for SI Calculation
Formula 1: “R as number”
The Simple Interest Formula is given by S.I = (P*N*R)/100.

Formula 2: “R as percentage”
The Simple-Interest Formula is given by S.I = P*N*R.

This shows how to calculate Simple Interest using the interest for the given principal amount P, with rate of interest R/period of a given period of time P.

Example of SI Calculation
If a principal amount of Rupees 1000 is borrowed/deposited by a person for a period of 2 years with 3% rate of interest, then the simple-interest is given by

By applying the values of P, N and R given, we get

Formula 1: R as number

We know that the rate of interest is 3%. We take R = 3

S. I.  = (P * N* R)/100
S. I.  = (1000 * 2 * 3)/100 = 6000/100 = 60

Formula 2: R as percentage

We know that R = 3% = 3/100 = .03

S.I. = P * N* R
S.I. = 1000*2*.03 = 60.00 = 60

Applications of SI
In post offices, schemes such as MIS pay simple interest for the amount deposited for five years as recurring deposit.

Credit cards charging simple-interest for the amount to be paid is more beneficial.

Parallelogram Vector addition

One of the most common vector operations that is frequently encountered is addition operation. Addition of two or more vectors to arrive to a vector sum is called vector addition. Consider two vectors,  u=(u1, u2) and  v=( v1, v2), the sum of these vectors would be vector u + vector v= (u1+v1, u2+v2). This sum is called the resultant vector. There are various methods to find the resultant vector namely, parallelogram method, component method, graphical method, cosine method, polygon method etc.  

Parallelogram Vector addition:  In this method first two vectors are drawn such that their initial points coincide. Then the other two lines are drawn to form a parallelogram. The resultant would be the diagonal of the parallelogram drawn from the initial point to the opposite vertex of the parallelogram.

Vector addition component method is one way used in adding vectors. Component means ‘part ’and hence they can be considered as the coordinates of the point that is associated with the vector. In a Euclidean plane consider two vectors, u=(u1, u2) and v=( v1, v2), the resultant vector which is the sum of these vectors is given by, u+v = (u1+v1, u2+v2). In a three dimensional space, given vectors u=(u1, u2, u3) and v=( v1, v2,v3) the method would be similar to the method used in addition of vectors in a Euclidean plane. So, u+v = (u1+v1, u2+v2, u3+v3). We can finally conclude that vector-addition is just like the normal addition, component by component.

Let us now learn the vector addition graphical method, consider two vectors, u=(4,3) and v=(1,4) in the plane. Using the component method of vector-addition the sum can be given as, u+v = (4+1, 3+4) = (5,7). Using the graphical method we get the same resultant vector by taking one vector whose direction and magnitude is unchanged and placing its end at the unchanged vector’s tip, and joining the origin and the new location of the displaced vector using an arrow. This procedure in general works for addition of vectors. For any two given vectors u and v in the plane, the sum of the vectors in general can be graphically represented as the vector addition diagram given below

As the vectors in a two dimensional space lie in the same plane, any two vectors in a three dimensional space also lie in the same plane and hence graphical method works well for vector-addition in a 3-dimensional space.

Understanding the Concept of Linear Combination

In mathematics the equations are very important. The problems can be converted into equations and then the equations can be solved with the help of multiple methods. The solutions obtained must be checked for their feasibility. This is because all the solutions obtained will not be feasible. Only the solutions that are feasible must be selected, otherwise the answer might go wrong. One must be very careful in selecting the solutions of the equations.

In case of linearity the degree of the given equation is ‘1’. The linear combination can be formed with the help of an equation. There need to be constants for this process to be performed. It also involves the simple process of addition. Basically it is the formation of an expression. This expression can be formed with the help of constants and the simple addition process.

The term in the expression must be multiplied with a constant and the answers obtained must be added. This will give the required solution. The linear combinations can be very helpful and can have various applications in mathematics.

The process of solving linear combinations must be learnt thoroughly to appreciate the concept. The examples will help in better explaining the concept. This concept is related to the concept of linear algebra. It is a branch of algebra and has to be learnt to understand this concept. This concept is also very helpful in vector theory and is a part of vector theory.

Vectors are different from scalar quantities. Both are quantities which have magnitude. The difference between vector and scalar quantities is that the vector quantities have a direction attached to them. But the scalar quantities do not have a direction attached to them.

This is the basic difference between the two quantities. So, one has to be very careful in dealing with these types of quantities. The direction can play a very important role. If the direction is not properly denoted the answer can go wrong. So, in the case of vector quantities, the direction plays a very important role. But in case of scalar quantities the magnitude plays the most important role.

The change in magnitude can make the whole answer go wrong in the case of scalar quantities. But in the case of vector quantities both magnitude and direction are important. Even if one goes wrong the whole answer goes wrong.