linear algebra sample final

Linear algebra is mainly use to find the solution of systems of linear equations in some unknowns. Linear algebra has a demonstration in analytic geometry and is generalize in operator theory. Linear algebra associates with the families of vectors called vector spaces, and with functions contain one input vector and output another vector, according to rules. Linear algebra has the application with the theory of systems of linear equations, determinants, and matrices. The sample problems are discussed below for final review.

Linear Algebra Sample Problems for Final:

The sample problems are solved in detail for final review as shown below.

Problem 1:

Evaluate the linear equation   -3(x - 4) / 6 - (x - 2) / 3 = -2 x

Sol:

The given equation has rational expressions. To eliminate the denominators by multiplying by the LCD
-3(x - 4) / 6 - (x - 2) / 3 = -2 x

The LCD is equal to 6*3 = 18. Multiply both sides by LCD.
18 * [-3 (x - 4) / 6 - (x - 2) / 3] = 18* [-2 x]

Simplify to eliminate the denominator.
-9(x - 4) - 6(x - 2) = -36x

Multiply factors and grouping the terms

-15x + 48 = -36x

Subtract 48 to both sides
-15x + 48 - 48 = -36x -48

Group like terms
-15x = -36x - 48

Add 36x to both sides
-13x + 36x = -36x - 48 +36x

Grouping the terms
23x = -48

Multiply both sides by `1/23 `
x = - `48/23`

Conclusion:
x = - `48/23` is the solution for the above given equation.

Sample Linear Algebra Practice Problems for Final:

1) Evaluate the linear equation   -3(x - 4) / 7 - (x - 2) / 2 = -2 x

2) Solve the linear equation -5(-x - 7) = 5x – 32.

3) Evaluate the linear equation   -3(-x +3) = x - 7

Answers:

1)   X = -38/15 is the solution for the above given equation.

2) The above equation has no solution.

3)  X=One is the solution for the above equation.

Order of Operations

Order or operations is one of the most important topics in mathematics. Order of operations is also known as PEMDAS or BODMAS. As the term suggests, order of operations refers to the steps to perform a mathematical operation. When it is a simple addition or subtraction, you can add or subtract instantly. For example: Sunny bought Nerf guns India from Fisher Price India collection. He already has Angry Birds toys India and Dartboard games.

How many variety of toys and games collection he has in total: Nerf guns + Angry Birds toys India + Dartboard = 3 variety of toys and games. This is addition operation in mathematics. John has three Nerf guns of Fisher Price India brand. He gave one to his cousin Andrew. How many Nerf guns does he have? 3 Nerf guns – 1 Nerf gun = 2 Nerf gun. This is subtraction operation in mathematics. But what do you do when a mathematical expression has more than one operation. For example: Sunny had 3 guns, his father gave him 2 new guns and he gave one to his friend John. How many he has now? Mathematically: (3+2) – 1 = 4. This is a mathematical expression.

While calculating a mathematical expression, it is essential to follow some rules or order of operation. This order of operation is referred to as BODMAS or PEMDAS as abbreviation for the functions in mathematics. The order of operations is expressed as below.
1. [B] Brackets or Parenthesis
2. [O] Orders (Exponents, roots etc)
3. [DM] Division and Multiplication
4. [AS] Addition and Subtraction

The above order of operation states that in a mathematical expression, at first the operation inside brackets needs to be calculated, thereafter the powers, roots and more, then division and multiplication and finally addition and subtraction. This should be the order of operation while simplifying or calculating a mathematical expression.
For example: There are six kids. Father bought one Nerf Guns India and two building block toys for each kid. How many toys father bought in total?
Mathematically, (1+2) x 6
Applying order of operations (1+2) = 3 x 6 = 18
Therefore, father bought 18 toys in total.

This is referred as order of operations or BODMAS in mathematics.

Estimating with Confidence

Concept of ‘estimation with confidence’ is normally used by everybody in their daily lives. For example, a person may say ‘I believe there are 90 percent chances that Congress will win in the next elections with more than 350 seats’. Here, the person’s confidence level is 90 percent and he is estimating by how much seats congress will win. Let’s move forward to the formal concept of ‘estimation with confidence’.

There are two types of tests one-tailed and two-tailed, we will discuss ‘estimation with confidence’ in each of the case separately.

Estimating with Confidence in a One Tailed Test

In the single tailed test we estimate with the given level of confidence whether the estimated value is same as the population value or either statistically greater than or less than the population value/expected value (for simplicity we are assuming normal distribution). In this we can have right or left tailed test.

Left Tailed : Let us assume the given level of confidence is 85%, from Fig-I we see yellow area is the rejection region. So, with 85% confidence we can say that our estimated value won’t b statistically smaller from the population value/expected value and there is only 15 % probability that it will be statistically smaller.

FIG-I

Right Tailed: In this case the calculation remains the same, the only difference we don’t want statistically greater value than population value/expected value rather than less than as in the previous case. We represent right tailed estimation as in Fig-II. Here yellow shaded region is the rejection area. Confidence level is 85%. So, with 85% confidence we can say that our estimated value won’t b statistically greater from the population value/expected value and there is only 15 % probability that it will be statistically greater.

FIG-II

Estimating with Confidence in Two Tailed Test

Under a two tailed estimation with confidence we do not want either too big or too small value in comparison to the population value/expected value. In other words, we have rejection area on the both tails of the curve (as shown in Fig-III). Here, the yellow shaded area is the rejection area. Here, the confidence level is of 70%. So, with 70% confidence we can say that our estimated value won’t b statistically different from the population value/expected value and there is only 30% probability that it will be statistically different.

FIG-III

Compare and Order Integers

A number line is good place to start when describing integers. As integers are all the counting numbers so starting from 0, the numbers to the right side of zero are called positive integers, and from 0 to the left side, all the numbers are called the negative integers.

Comparing Integers - We can Compare Integers with their Opposites which means the numbers on the opposite sides with equal distance from zero on the number line. For example, on a number line, 4 and -4 are the opposites. The number zero is the opposite of itself, and it is considered neither positive nor negative. Let us graph the integers and their opposites. Let’s graph 4 and its opposite on the number line. Firstly graph 4 by placing a dot at positive 4 on the number line. We know the opposite of positive 4 is negative 4. So place a dot at a negative 4 on the number line to complete the problem. Next example lets graph -2 and its opposite on the number line. First, graph negative 1 by placing a dot on the negative on the number line. Next step we know the opposite of negative 1 is positive is 1. So we place a dot on number line to complete our graph.

Ordering Integers – Ordering integers means Ordering Integers from least to Greatest and for this we write integers in the ascending order. The numbers on left of number line is smaller than the number on the right. Comparing and Ordering Integers can be done by plotting them on the number line. For example: - If we have numbers -3, 0, 5,-5,-1 and 4 and we need Compare and Order Integers, we first plot them on the number line and see the integers to the left on the number line are smaller to those on the right. So reading the dots on the number line from left to right gives us the order from least to the greatest. Here we have to do is read the dots from left to right and we have our order. The smallest number is here on the extreme left will be -5 and then it’s -3, then -1, then 0, then 4 and then 5 which is on extreme right. So we can write the order as -5 < -3 < -1 < 0 < 4 < 5.

Numerator and Denominator

Numerator and Denominator Definition - In mathematics, when we study fractions, the numerator and denominator are very important words. In a fraction, the top part is known as the numerator and the bottom part is termed as the denominator.  To understand the concept better, let us draw a circle that is divided into four equal parts, we call them fourths where one of the parts has been shaded. Then we say that one fourth of the circle has been shaded and it is written as ¼, here 4 is the denominator and 1 is the numerator. In Fractions Numerator and Denominator are represented like this Numerator/Denominator.

How do we Define Numerator and Denominator- The denominator describes the number of parts, and the numerator describes the number of parts that are shaded.

How to Remember Numerator and Denominator - Where do these names come from and how do they make sense? Denominator means the name. It is the name for the fraction. Well there is a history to the word denominator, which helps to understand the connections. When we go to the church we might say it has a certain denominations, Luther is one denomination, catholic is another denomination. There is another use of word denomination, when people have bills, a dollar bill or five dollar bill, those are different denominations, or the name that they have.

In school election, we might have noticed, that we nominate somebody to be the president if require. It is a similar word nominate. Nominate means to name somebody to be the president. Or in Spanish, the word nombre that means the name, so nombre, nominate, denomination they all relate to the word denominator. They come from the same land root of denomino.

So denominator is just a name. Whatdoes numerator mean? The number of parts that we talking about. So we have drawn four equal parts, how many shaded parts is the number that represents a numerator. Let us understand it by using an example that is identifying the numerator and denominator in the fraction 3/4.

The numerator is the number on the top in fraction. Here we have ¾ where 3 is on the top so 3 is the numerator and the denominator is the number in the bottom of the fractions. It is 4 here. This fraction represents 3 out of 4 pieces of a pie. The denominator shows total number of pieces of pie and 3 is represented as the part of those 4 pieces.

Definition of Mean

It is also called as arithmetic average, or simply average. I is defined as the arithmetic average of the set of data value and is calculated by the sum of all the values from the data and divide by size of the data. Especially the term arithmetic average is used in mathematics and statics, since it help us to differentiate it from other methods such as harmonic and geometric average. It is most frequently used in mathematics and statistics fields such as economics, sociology and history.

It is generally sample average or population average. If the set of observation is a statistical sample, then that average is called as sample average. If the set of observation is a statistical population, then that mean is called as population average.  The sample average is denoted by the symbol ( ) which also called as x bar and the population average is denoted by the symbol of Greek letter (µ). Both the sample and population average is calculated by adding all the observations and divided by the number of observations.

Formula for Mean
Formula for Sample average
= (∑¦x)/n
Where,
- Sample average
Sx - Sum of all the sample observation
n - Number of sample observation
Formula for Population average formula
µ =  (∑¦x)/N
Where,
µ - Population average
SX - Sum of all population observation
N - Number of population observation


Define Mean
It is defined as it the average of numbers from given set of data and is calculated by just addition of all numbers from set of data and divides them by the size of the data. Simply it is a just average of the given numbers. Average is represented by the symbol .
  = (∑¦x)/n
Where,
- Mean
S - Symbol of summation
x - Values present in given set of data
n - Number of data or size of data or how many numbers present in set of data

Definition of Mean
It is an arithmetic average of set of a data values and is not to be a middle value or any one value from the set of data values. It is a central value of a set of numbers or data. Calculation of average is very simple. Averages is calculated by adding all the numbers and divide them by how many numbers there are, in other words, it is the sum of the numbers divided by the count of numbers.
For example to find the average for given set of data such as 6, 11, 7, 4. To calculate average first add all the numbers,
6 + 11 + 7 + 4 = 28
Then dived the resulting number by how many numbers are their in data set, here there are 4 numbers are their, so
28/4 = 7
The average value is 7.

Sample Mean Formula
Where Sn = x1, x2, x3,………xn random sample of size n, then the formula for sample average is,
= (∑¦x)/n
Where,
- Sample average
S - Symbol of summation
x - Values present in the sample set of data
n - Number of sample data

How to find Directional Derivatives?

In calculus, the directional derivative is a differentiable function which multivariate along a vector V with a point P. It represents nothing but the rate of change of a function which moves in the direction of V, through the point P. The directional derivatives are the generalization of the partial derivatives; where in one of the coordinate axis is always parallel in direction.

Mathematical Definition of Directional Derivatives
The directional derivative is defined as the rate of change of the scalar function f(x) = f(x1, x2, x3... xn) along the unit vector u = (u1, u2, u3….. un), in which the unit vector is defined by a limit in terms of h. Therefore the mathematical definition for the directional derivative can be written as,

 Du f(x) = lim h->0+ f(x+hu) – f(x) / h.

Derivation of Directional derivative

It is simple method to find the directional derivative formula which will be used to simplify the problems. Let us define a function that is made up of a single variable, such as g(z) which equals f(sum of x0 + az and y0 + bz). Here x0, y0, a, b are some numbers which are fixed. Now, z is the letter which does not represent a fixed number.  Then, by the actual definition, for a single variable function we know that, g’(z) equals the limit which tends to h->0 of g (sum of z and h) – g(z) whole divided by h.  Now applying the value of the derivative z as zero, we will get, g’(0) which equals limit which tends to h->0 of subtraction of g (h) and g(0) whole divided by h. Substituting the equation of g(z) mentioned earlier we will get g’(0) as lim h->0 of f(x0 + az, y0 + bz) – f(x0,y0) whole divided by h. Then this equation will be obviously equal to the Du f(x0, y0). Therefore we will finally come with a relationship of g’(0) which equals Du f(x0, y0). Now we can rewrite g(z) as g(z) = f(x, y) where x will be equal to x0 + az and y will be equal to y0 + bz. Then, applying the chain rule we will get the equation as,  g’(z) = dg/dz = Sum of (multiplication of df/dx and dx/dz ) and (multiplication of df/dy and dy/dz), which will be equal to sum of fx (x,y) a and  fy (x,y).After taking z=0, we will get x as x0 and y as y0 and applying these, we will get, g’(0) as sum of            fx (x0,y0) a and fy (x0,y0) b. Now equating this equation with the g’(z) equation of very first we derived, we will arrive with a formula as shown below:

Du f(x, y) = fx (x, y) a + fy (x, y) b.

This equation will provide a simple way of calculation than the previous limit based definition.
Directional derivatives are used in solving the matrix symmetrical valued functions; especially the second directional derivative and it plays a vital role in solving non linear type problems.