Algebra for dummies pdf free download

If you're vexed by variables, Algebra I For Dummies, 2nd Edition provides the plain-English, easy-to-follow guidance you need to get the right solution every time!

You'll understand how to factor fearlessly, conquer the quadratic formula, and solve linear equations. It includes real-world examples and story problems that will help even the most entrenched algebra phobes appproach the subject with ease. There are no reviews yet. Be the first one to write a review. Books for People with Print Disabilities. Internet Archive Books. Scanned in China. The common theme of such designs is that they contain a set or grouping of objects that all have something in common.

Certain properties are attached to the plan — properties that apply to all the members of the grouping. Vector spaces contain vectors, which really take on many different forms. The easiest form to show you is an actual vector, but the vectors may actu- ally be matrices or polynomials. As long as these different forms follow the rules, then you have a vector space.

In Chapter 14, you see the rules when investigating the subspaces of vector spaces. The rules regulating a vector space are highly dependent on the operations that belong to that vector space.

You find some new twists to some famil- iar operation notation. With vector spaces, the operation of addition may be defined in a completely different way. Does that rule work in a vector space? Determining Values with Determinants A determinant is tied to a matrix, as you see in Chapter The determi- nant incorporates all the elements of a matrix into its grand plan.

You have a few qualifications to meet, though, before performing the operation determinant. Square matrices are the only candidates for having a determinant.

Let me show you just a few examples of matrices and their determinants. The determinants of the respective matrices go from complicated to simple to compute. For example, the matrix D, that I show you here, has a determinant of 0 and, consequently, no inverse. Matrix D looks perfectly respectable on the surface, but, lurking beneath the surface, you have what could be a big problem when using the matrix to solve problems. You need to be aware of the consequences of the determi- nant being 0 and make arrangements or adjustments that allow you to pro- ceed with the solution.

The values of the variables are ratios of dif- ferent determinants computed from the coefficients in the equations. Zeroing In on Eigenvalues and Eigenvectors In Chapter 16, you see how eigenvalues and eigenvectors correspond to one another in terms of a particular matrix.

Each eigenvalue has its related eigen- vector. So what are these eigen-things? An eigen- value is a number, called a scalar in this linear algebra setting. For example, let me reach into the air and pluck out the number For now, just trust me on this. The resulting vector is the same whether I multiply the vector by 13 or by the matrix. You can find the hocus-pocus needed to do the multiplication in Chapter 3.

I just want to make a point here: Sometimes you can find a single number that will do the same job as a complete matrix. I actually peeked. Every matrix has its own set of eigenvalues the numbers and eigenvectors that get multiplied by the eigenvalues. In Chapter 16, you see the full treatment — all the steps and procedures needed to discover these elusive entities.

A vector is a special type of matrix rectangular array of numbers. The vectors in this chapter are columns of numbers with brack- ets surrounding them. Two-space and three-space vectors are drawn on two axes and three axes to illustrate many of the properties, measurements, and operations involving vectors. You may find the discussion of vectors to be both limiting and expanding — at the same time.

Vectors seem limiting, because of the restrictive structure. As with any mathematical presentation, you find very specific meanings for otherwise everyday words and some not so everyday. Keep track of the words and their meanings, and the whole picture will make sense. Lose track of a word, and you can fall back to the glossary or italicized definition. Describing Vectors in the Plane A vector is an ordered collection of numbers. Vectors containing two or three numbers are often represented by rays a line segment with an arrow on one end and a point on the other end.

Representing vectors as rays works with two or three numbers, but the ray loses its meaning when you deal with larger vectors and numbers. The properties that apply to smaller vectors also apply to larger vectors, so I introduce you to the vectors that have pictures to help make sense of the entire set.

When you create a vector, you write the numbers in a column surrounded by brackets. Vectors have names no, not John Henry or William Jacob.

The names of vectors are usually written as single, boldfaced, lowercase letters. You often see just one letter used for several vectors when the vectors are related to one another, and subscripts attached to distinguish one vector from another: u1, u2, u3, and so on. Here, I show you four of my favorite vectors, named u, v, w, and x: The size of a vector is determined by its rows or how many numbers it has.

Technically, a vector is a column matrix matrices are covered in great detail in Chapter 3 , meaning that you have just one column and a certain number of rows. Vectors in two-space are represented on the coordinate x,y plane by rays. In standard position, the ray representing a vector has its endpoint at the origin and its terminal point or arrow at the x,y coordinates designated by the column vector. The x coordinate is in the first row of the vector, and the y coordinate is in the second row.

The coordinate axes are used, with the horizontal x-axis and ver- tical y-axis. Figure shows the six vectors listed in the preceding section, drawn in their standard positions.

The coordinates of the terminal points are indicated on the graph. The following vector is just as correctly drawn by starting with the point —1,4 as an endpoint, and then drawing the vector by moving two units to the right and three units down, ending up with the terminal point at 1,1.

Both the length and the direction uniquely determine a vector and allow you to tell if one vector is equal to another vector. Vectors can actually have any number of rows. Also, the applica- tions for vectors involving hundreds of entries are rather limited and difficult to work with, except on computers. Adding a dimension with vectors out in space Vectors in R3 are said to be in three-space. The vectors representing three- space are column matrices with three entries or numbers in them.

The R part of R3 indicates that the vector involves real numbers. Three-space vectors are represented by three-dimensional figures and arrows pointing to positions in space. Picture a vector drawn in three- space as being a diagonal drawn from one corner of a box to the opposite corner.

A ray representing the following vector is shown in Figure with the endpoint at the origin and the terminal point at 2,3,4. Vectors are groupings of numbers just waiting to have operations performed on them — ending up with predictable results. The different geometric transformations performed on vectors include rota- tions, reflections, expansions, and contractions. You find the rotations and reflections in Chapter 8, where larger matrices are also found.

As far as oper- ations on vectors, you add vectors together, subtract them, find their oppo- site, or multiply by a scalar constant number. You can also find an inner product — multiplying each of the respective elements together. Swooping in on scalar multiplication Scalar multiplication is one of the two basic operations performed on vec- tors that preserves the original format. You may not be all that startled by this revelation, but you really should appreciate the fact that the scalar main- tains its original dimension.

Reading the recipe for multiplying by a scalar A scalar is a real number — a constant value. Multiplying a vector by a scalar means that you multiply each element in the vector by the same constant value that appears just outside of and in front of the vector.

Chapter 2: The Value of Involving Vectors 25 Opening your eyes to dilation and contraction of vectors Vectors have operations that cause dilations expansions and contractions shrinkages of the original vector.

Both operations of dilation and contrac- tion are accomplished by multiplying the elements in the vector by a scalar. If the scalar, k, that is multiplying a vector is greater than 1, then the result is a dilation of the original vector.

If the scalar, k, is a number between 0 and 1, then the result is a contraction of the original vector. In Figure , you see the results of the dilation and contraction on the origi- nal vector. You also may have wondered why I only multiplied by numbers greater than 0. The rule for contractions of vectors involves numbers between 0 and 1, nothing smaller. In the next section, I pursue the negative numbers and 0.

The illustration for multiplying by 0 in two-space is a single point or dot. Not unexpected. The zero vector is the identity for vector addition, just as the number 0 is the identity for the addition of real numbers. When you multiply a vector by —2, as shown with the following vector, each element in the vector changes and has a greater absolute value: In Figure , you see the original vector as a diagonal in a box moving upward and away from the page and the resulting vector in a larger box moving downward and toward you.

Chapter 2: The Value of Involving Vectors 27 z —2,3,5 5 —6 y 3 —10 Figure Multiplying x a vector by 4,—6,—10 a negative scalar. Adding and subtracting vectors Vectors are added to one another and subtracted from one another with just one stipulation: The vectors have to be the same size.

The process of adding or subtracting vectors involves adding or subtracting the corresponding elements in the vectors, so you need to have a one-to-one match-up for the operations. Figure shows all three vectors. So, if you want to change a subtraction problem to an addition problem perhaps to change the order of the vectors in the operation , you rewrite the second vector in the problem in terms of its opposite. For example, changing the following subtraction problem to an addition problem, and rewriting the order, you have: Yes, of course the answers come out the same whether you subtract or change the second vector to its opposite.

The maneuvers shown here are for the structure or order of the problem and are used in various applications of vectors. Vectors with more than three rows also have magnitude, and the computation is the same no matter what the size of the vector. The magnitude of vector v is designated with two sets of vertical lines, v , and the formula for computing the magnitude is where v1, v2 ,.

The box measures 3 x 2 x 4 feet. How long a rod can you fit in the box, diagonally? According to the formula for the magnitude of the vector whose numbers are the dimensions of the box, you can place a rod measuring about 5. Adjusting magnitude for scalar multiplication The magnitude of a vector is determined by squaring each element in the vector, finding the sum of the squares, and then computing the square root of that sum.

What happens to the magnitude of a vector, though, if you mul- tiply it by a scalar? Can you predict the magnitude of the new vector without going through all the computation if you have the magnitude of the original vector?

Chapter 2: The Value of Involving Vectors 31 The magnitude of the new vector is three times that of the original. So it looks like all you have to do is multiply the original magnitude by the scalar to get the new magnitude.

Careful there! In mathematics, you need to be suspicious of results where someone gives you a bunch of numbers and declares that, because one example works, they all do. So, if you multiply a vector by a negative number, the value of the magnitude of the resulting vector is still going to be a positive number. Making it all right with the triangle inequality When dealing with the addition of vectors, a property arises involving the sum of the vectors.

The theorem involving vectors, their magnitudes, and the sum of their magnitudes is called the triangle inequality or the Cauchy- Schwarz inequality named for the mathematicians responsible.

For any vectors u and v, the following, which says that the magnitude of the sum of vectors is always less than or equal to the sum of the magnitudes of the vectors, holds: Showing the inequality for what it is In Figure , you see two vectors, u and v, with terminal points x1,y1 and x2,y2 , respectively.

The triangle inequality theorem says that the magnitude of the vector resulting from adding two vectors together is either smaller or sometimes the same as the sum of the magnitudes of the two vectors being added together. Then I compare the magni- tude to the sum of the two separate magnitudes.

The sums are mighty close, but the magnitude of the sum is smaller, as expected. You find the average of two numbers by adding them together and dividing by two.

To find a geometric mean of two numbers, you just determine the square root of the product of the numbers. The geometric mean of a and b is while the arithmetic mean is For an example of how the arithmetic and geometric means of two numbers compare, consider the two numbers 16 and The geometric mean is the square root of the product of the numbers.

In this example, the geometric mean is slightly smaller than the arithmetic mean. In fact, the geometric mean is never larger than the arithmetic mean — the geometric mean is always smaller than, or the same as, the arithmetic mean. I show you why this is so by using two very carefully selected vectors, u and v, which have elements that illustrate my statement.

First, let Assume, also, that both a and b are positive numbers. To get to the last step, I used the commutative property of addition on the left changing the order and found that I had two of the same term. Now I square both sides of the inequality, divide each side by 2, square the binomial, distribute the 2, and simplify by subtracting a and b from each side: See!

The geometric mean of the two numbers, a and b, is less than or equal to the arithmetic mean of the same two numbers. Getting an inside scoop with the inner product The inner product of two vectors is also called its dot product. His birth, during a time of At one point, Cauchy responded to a request political upheaval, seemed to set the tone for from the then-deposed king, Charles X, to tutor the rest of his life. His family was often visited by mathematicians Cauchy was raised in a political environment of the day — notably Joseph Louis Lagrange and was rather political and opinionated.

He and Pierre-Simon Laplace — who encouraged was, more often than not, rather difficult in the young prodigy to be exposed to languages, his dealings with other mathematicians.

No first, and then mathematics. Cauchy was quick to publish his find- abandoned engineering for mathematics. At ings unlike some mathematicians, who tended a time when most jobs or positions for math- to sit on their discoveries , perhaps because of ematicians were as professors at universities, an advantage that he had as far as getting his Cauchy found it difficult to find such a position work in print.

He was married to Aloise de Bure, because of his outspoken religious and political the close relative of a publisher. The superscript T in the notation uTv means to transpose the vector u, to change its orientation. The reason should become crystal clear in the following section.

Consider the two vectors shown in Figure , which are drawn perpendicular to one another and form a degree angle or right angle where their endpoints meet.

You can confirm that the rays forming the vectors are perpendicular to one another, using some basic algebra, because their slopes are negative reciprocals. The slope of a line is determined by finding the difference between the y-coor- dinates of two points on the line and dividing that difference by the difference between the corresponding x-coordinates of the points on that line.

And, further, two lines are perpendicular form a right angle if the product of their slopes is —1. So what does this have to do with vectors and their orthogonality? Read on. If the inner product of vectors u and v is equal to 0, then the vectors are perpendicular.

Referring to the two vectors in Figure , you have Now, finding their inner product, Since the inner product is equal to 0, the rays must be perpendicular to one another and form a right angle. In Figure , you see the two vectors whose terminal points are 2,6 and —1,5. Then put the numbers in their respective places in the formula: Using either a calculator or table of trigonometric functions, you find that the angle whose cosine is closest to 0.

The angle formed by the two vectors is close to a degree angle. Matrices have their own arithmetic. What you think of when you hear multiplication has just a slight resemblance to matrix mul- tiplication. Matrix algebra has identities, inverses, and operations.

Getting Down and Dirty with Matrix Basics A matrix is made up of some rows and columns of numbers — a rectangular array of numbers. You have the same number of numbers in each row and the same number of numbers in each column.

The number of rows and col- umns in a matrix does not have to be the same. A vector is a matrix with just one column and one or more rows; a vector is also called a column vector. Matrices are generally named so you can distinguish one matrix from another in a discussion or text.

Nice, simple, capital letters are usually the names of choice for matrices: Matrix A has two rows and two columns, and Matrix B has four rows and six columns. The rectangular arrays of numbers are surrounded by a bracket to indicate that this is a mathematical structure called a matrix. The different positions or values in a matrix are called elements. The elements themselves are named with lowercase letters with subscripts. The subscripts are the index of the element.

The element a12 is in matrix A and is the number in the first row and second column. A gen- eral notation for the elements in a matrix A is aij where i represents the row and j represents the column. In matrix B, you refer to the elements with bij. Sometimes a rule or pattern is used to construct a particular matrix.

Defining dimension Matrices come in all sizes or dimensions. The dimension gives the number of rows, followed by a multiplication sign, followed by the number of columns. Determining the dimension of a matrix is important when performing opera- tions involving more than one matrix. When adding or subtracting matrices, the two matrices need to have the same dimension. When multiplying matri- ces, the number of columns in the first matrix has to match the number of columns in the second matrix.

You find more on adding, subtracting, multi- plying, dividing, and finding inverses of matrices later in this chapter. And each operation requires paying attention to dimension.

Putting Matrix Operations on the Schedule Matrix operations are special operations defined specifically for matrices. When you do matrix addition, you use the traditional process of addition of numbers, but the operation has special requirements and specific rules. Matrix multiplication is actually a combination of multiplication and addition. Adding and subtracting matrices Adding and subtracting matrices requires that the two matrices involved have the same dimension.

The matrices rectangular arrangements always have the same type of policy in each column and the same agents in each row. The rectangular array allows the sales manager to quickly observe any trends or patterns or problems with the production of the salespersons.

Matrix addition is commutative. Matrix subtraction, however, is not commutative. Chapter 3: Mastering Matrices and Matrix Algebra 45 Scaling the heights with scalar multiplication Multiplying two matrices together takes some doing — perhaps like climbing the Matterhorn. But scalar multiplication is a piece of cake — more like riding the tram to the top of yonder hill. I just wanted to set you straight before proceeding. Multiplying a matrix A by a scalar constant number , k, means to multiply every element in matrix A by the number k out in front of the matrix.

So, multiplying some matrix A by —4, Making matrix multiplication work Matrix multiplication actually involves two different operations: multiplica- tion and addition. Elements in the respective matrices are aligned carefully, multiplied, added, and then the grand sum is placed carefully into the result- ing matrix. Matrix multiplication is only performed when the two matrices involved meet very specific standards.

Multiplying two matrices The process used when multiplying two matrices together is to add up a bunch of products. Each element in the new matrix created by matrix mul- tiplication is the sum of all the products of the elements in a row of the first matrix times a column in the second matrix.

Let me show you an example before giving the rule symbolically. The number of columns in matrix K is 3, as is the number of rows in matrix L. Even when you have two square matrices the number of rows and number of columns are the same , their product usually is not the same when the matrices are reversed. Having said that, I have to tell you that there are cases where matrix multiplication is commutative.

Putting Labels to the Types of Matrices Matrices are traditionally named using capital letters. So you have matrices A, B, C, and so on. Matrices are also identified by their structure or elements; you identity matrices by their characteristics just as you identify people by their height or age or country of origin.

Matrices can be square, identity, tri- angular, singular — or not. Chapter 3: Mastering Matrices and Matrix Algebra 49 Identifying with identity matrices The two different types of identity matrices are somewhat related to the two identity numbers in arithmetic.

The additive identity in arithmetic is 0. The same idea works for the multiplicative identity: The multi- plicative identity in arithmetic is 1. You multiply any number by 1, and the number keeps its original identity. Zeroing in on the additive identity The additive identity for matrices is the zero matrix. A zero matrix has ele- ments that are all zero. How convenient! But the zero matrix takes on many shapes and sizes.

Matrices are added together only when they have the same dimen- sion. When adding numbers in arithmetic, you have just one 0. But in matrix addition, you have more than one 0 — in fact, you have an infinite number of them technically. Sorta neat. In addition to having many additive identities — one for each size matrix — you also have commutativity of addition when using the zero matrix.

Addition is commutative, anyway, so extending commutativity to the zero matrix should come as no surprise. The common trait of the multi- plicative identity is that the multiplicative identity also comes in many sizes; the difference is that the multiplicative identity comes in only one shape: a square. The multiplicative identity is a square matrix, and the elements on the main diagonal running from the top left to the bottom right are 1s.

All the rest of the elements in the matrix are 0s. When you multiply a matrix times an identity matrix, the original matrix stays the same — it keeps its identity. Of course, you have to have the correct match-up of columns and rows. For example, let me show you matrix D being multiplied by identity matrices. The size of the identity matrix is pretty much dictated by the dimension of the matrix being multiplied and the order of the multiplication.

The exception to that rule is when a square matrix is multiplied by its identity matrix. You have commutativity of multiplication in this special case. A triangular matrix is either upper triangu- lar or lower triangular. The best way to define or describe these matrices is to show you what they look like, first. Matrix A is an upper triangular matrix; all the elements below the main diago- nal the diagonal running from upper left to lower right are 0s.

Matrix B is a lower triangular matrix; all the elements above the main diagonal are 0s. And matrix C is a diagonal matrix, because all the entries above and below the main diagonal are 0s. Triangular and diagonal matrices are desirable and sought-after in matrix applications. Doubling it up with singular and non-singular matrices The classification as singular or non-singular matrices applies to just square matrices.

Square matrices get quite a workout in linear algebra, and this is just another example. A square matrix is singular if it has a multiplicative inverse; a matrix is non-singular if it does not have a multiplicative inverse. When a matrix has a multiplicative inverse, the product of the matrix and its inverse is equal to an identity matrix multiplicative identity.

And, further- more, you can multiply the two matrices involved in either order commu- tativity and still get the identity. Connecting It All with Matrix Algebra Arithmetic and matrix algebra have many similarities and many differences. To begin with, the components in arithmetic and matrix algebra are com- pletely different. In arithmetic, you have numbers like 4, 7, and 0. In matrix algebra, you have rectangular arrays of numbers surrounded by a bracket. In this section, I get down-and-dirty and discuss all the properties you find when working with matrix algebra.

You need to know if a particular operation or property applies so that you can take advantage of the property when doing computations. The text definitely covers all topics that are covered in a usual College Algebra class, and actually it covers much more.

The extensive coverage of Systems of Equations and Matrices can not be really squeezed into a one semester College The version of the text we were provided had the trigonometry chapters cut out This was done simply by clipping the pdf rather than recompiling the latex, so the table of contents and index still reflect the full text, which is silly and If you're vexed by variables, Algebra I For Dummies, 2nd Edition provides the plain-English, easy-to-follow guidance you need to get the right solution every time!

You'll understand how to factor fearlessly, conquer the quadratic formula, and solve linear equations. After doing any. Trigonometry is a br anch of mathematics involving the study of tr iangles, and has.