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The system is solved by creating an augmented matrix where each column of the matrix corresponds to one of the vectors in the vector equation.
The solutions correspond to the scalars needed for the vector equation. For example, consider the following set of vectors and the target vector, b: You want to solve the vector equation formed by multiplying the vectors in the set by scalars and setting them equal to the target vector. Creating the augmented matrix for the system of equations and solving for the scalars, you have: After performing some row operations, you find that the last row of the matrix has 0s and an 8.
The vector b is not one of the linear combinations possible from the chosen set of vectors. Chapter 5: Lining Up Linear Combinations 93 Searching for patterns in linear combinations Many different vectors can be written as linear combinations of a given set of vectors. Conversely, you can find a set of vectors to use in writing a particu- lar target vector. Finding a vector set for a target vector For example, if you want to create the vector you could use the set of vectors and the linear combination The vector set and linear combination shown here are in no way unique; you can find many different combinations and many different vector sets to use in creating the particular vector.
Note, though, that my set of vectors is somewhat special, because the elements are all either 0 or 1. You see more vectors with those two elements later in this chapter and in Chapters 6 and 7.
When a set of vectors is very large or even has an infinite number of members, a pattern and generalized rule is preferable to describe all those members, if this is possible. Consider the following vectors: One possibility for describing the vectors in this set is with a rule in terms of two real numbers, a and b, as shown here: Two elements, the first and third, determine the values of the other two elements.
This rule is just one possibility for a pattern in the set of vectors. When you have only a few vectors to work with, you have to proceed with caution before applying that pattern or rule to some specific application.
The rule shows how to construct the vectors using the one vector and its elements. An alternative to using the one vector is to use two vectors and a linear combination: So many choices! You see the two vectors and two points in Figure Refer to Chapter 2 for more on graphing vectors.
Linear combinations of vectors are represented using parallel lines drawn through the multiples of the points representing the vectors. Now I get to introduce you to the span of a set of vectors. But, on the other hand, a span can be finite if you choose to limit the range of the scalars being used. The concept or idea of span gives a structure to the various linear combina- tions of a set of vectors. The vectors in a set pretty much determine how wide or small the scope of the results is.
The set of all linear combinations of this set is called its span. That set of linear combinations of the vectors is spanned by the original set of vectors. With all the real number possibilities for the scalars, the resulting set of vectors is infinite. Broadening a span as wide as possible A span of a vector set is the set of all vectors that are produced from linear combinations of the original vector set. The most comprehensive or all- encompassing spans are those that include every possibility for a vector — all arrangements of real numbers for elements.
Showing which vectors belong in a span A span may be all-encompassing, or it may be very restrictive. To determine whether a vector is part of a span, you need to determine which linear com- bination of the vectors produces that target vector.
If you have many vectors to check, you may choose to determine the format for all the scalars used in the linear combinations. So you find that a linear combination of the vectors does produce the target vector. The vector b does belong in the span. Writing a general format for all scalars used in a span Solving a system of equations to determine how a vector belongs in a span is just fine — unless you have to repeat the process over and over again for a large number of vectors.
Another option to avoid repeating the process is to create a general format for all the vectors in a span. Now add this new equation to twice the third equation. A common question in applications of linear algebra is whether a particular set of vec- tors spans R2 or R3.
Seeking sets that span When a set of vectors spans R2, then you can create any possible point in the coordinate plane using linear combinations of that vector set. I show you that the statement is true by writing the linear com- bination and solving the corresponding system of linear equations. So any vector can be written as a linear combination of the two vectors in the set. So, for any vector in R3, Ferreting out the non-spanning sets The set of vectors does not span R2. The non-spanning set may seem more apparent to you because one of the vectors is a multiple of the other.
Any linear combination written with the vectors in the set has only the zero vector for a solution. The term Ax indicates that matrix A and vector x are being multiplied together. In Chapter 3, you find all sorts of information on matrices — from their char- acteristics to their operations. In Chapter 4, you see systems of equations and how to utilize some of the properties of matrices in their solutions.
And here, in Chapter 6, you find the vectors, taken from Chapter 3, and I introduce you to some of the unique properties associated with vectors — properties that lend themselves to solving the equation and its corresponding system.
In this chapter, everything is all rolled together into one happy, usually cohesive family of equations and solutions. You find tech- niques for determining the solutions when they exist , and you see how to write infinite solutions in a symbolically usable expression.
When preparing to perform multiplication involving matrices, you first take note of why you want to multiply them and then decide how. The simplest multiplication involving matrices is scalar multiplication, where each element in the matrix is multiplied by the same multiplier. As simple as scalar multiplication may appear, it still plays an important part in solving matrix equations and equations involving the products of matrices and vectors.
The second type of multiplication involving matrices is the more complicated type. You multiply two matrices together only when they have the correct matrix dimensions and correct multiplication order. Establishing a link with matrix products In Chapter 3, you find the techniques needed to multiply two matrices together. Revisiting matrix multiplication When multiplying the matrix A times the vector x, you still have the same multiplication rule applying: The number of columns in matrix A must match the number of rows in vector x.
Looking at systems Multiplying a matrix and a vector always results in a vector. The matrix multi- plication process is actually more quickly and easily performed by rewriting the multiplication problem as the sum of scalar multiplications — a linear combination, in fact. This is the commutative property of scalar multiplication. This is the distributive prop- erty of scalar multiplication. Now throw into the mix another vector, b, that has m rows.
The matrix equation is written first with A as a matrix of colums ai and then with the elements in vector x multiplying the columns of A.
When multiplied out, you see the results of the scalar multiplication. To determine whether you have a solution, use the tried-and-true method of creating an augmented matrix correspond- ing to the columns of A and the vector b and then performing row operations. For example, given the matrix A and vector b, I find the vector x that solves the equation. Refer to Chapter 3 and multiplication of matrices if you need more information on multiplying and dimensions. You can find the explanation in Chapter 4.
The numbers in the last column correspond to the single solution for the vector x. Making way for more than one solution Many matrix-vector equations have more than one solution. When determin- ing the solutions, you may try to list all the possible solutions or you may just list a rule for quickly determining some of the solutions in the set.
You use the augmented matrix and reduce that matrix to echelon form. The last row in the augmented matrix contains all 0s. Having a row of 0s indicates that you have more than one solution for the equation. Take the reduced form of the matrix and go back to a system of equa- tions format, with the reduced matrix multiplying vector x and setting the product equal to the last column.
Gauss lived from until But, with the help made contributions not only to mathematics, of his mother and a family friend, Gauss was but also to astronomy, electrostatics, and able to attend the university. Gauss was a child prodigy — to the Gauss is credited with developing the technique delight of his parents and, often, to the dismay Gaussian elimination, a method or procedure for of his teachers. One of the more well-known stories of his child- After Gauss died, his brain was preserved and hood has to do with a tired, harried teacher studied.
His brain was found to weigh 1, who wanted to have a few minutes of peace grams an average male brain weighs between and quiet. The brain also contained highly complete the task. Gauss had the answer and had stum- bled on the basics to the formula for finding the sum of n integers.
Do the matrix multiplication and write the corresponding system of equations. Now solve for x1 in the first equation, substituting in the equivalences of x2 and x3. What I show you here is the situation where matrix A has specific values, and vector b varies.
The vector b is different every time you fill in random numbers. Then, after you choose numbers to fill in for the elements in b, the solutions in vector x correspond to those numbers.
For example, consider matrix A and some random matrix b. Divide each term in row two by 12, and divide each term in row three by 5. Multiply row two by 4 and add it to row one. Once you pick some values for vector b, you also have the solution vector x. By convenient, I mean that the elements will change the fractions in the formulas to integers. So, if then and the matrix-vector equation reads But, the beauty of this matrix-vector equation is that any real numbers can be used for elements in vector b.
The equation is not equal to 0 for very many choices of b1, b2, and b3. Using the numbers for the last column and the original matrix A, I now perform row operations on the augmented matrix. Solving a matrix-vector equation seems like a simple enough situation: Just find some elements of a vector to multiply times a matrix so that you have a true statement. The last row in the matrix has three 0s and then a fraction.
The statement is impossible. When you multiply by 0, you always get a 0, not some other number. So the matrix-vector equation has no solution. Expanding your search in hopes of a solution Consider the matrix C and vector d. Writing the augmented matrix and using row operations, the last row has zeros and the fractional expression.
The value of must be equal to 0. You find 0s set equal to sums of linear terms instead of nonzero numbers. Homogeneous systems of equations are set equal to 0, and then you try to find nonzero solutions — an interesting challenge. Also in this chapter, I tie linear independence with span to produce a whole new topic called basis. Old, familiar techniques are used to investigate the new ideas and rules. Unlike a general system of linear equations, a homogeneous system of linear equations always has at least one solution — a solution is guaranteed.
The guaranteed solution occurs when you let each variable be equal to 0. The solution where everything is 0 is called the trivial solution. The trivial solution has each element in the vector x equal to 0. Determining the difference between trivial and nontrivial solutions A system of linear equations, in general, may have one solution, many solu- tions, or no solution at all.
For a system of linear equations to have exactly one solution, the number of variables cannot exceed the number of equa- tions. A system of linear equations may have more than one solution. Many solu- tions occur when the number of equations is less than the number of variables. To identify the solutions, you assign one of the variables to be a parameter some real number and determine the values of the other vari- ables based on formulas developed from the relationships established in the equations.
In the case of a homogeneous system of equations, you always have at least one solution. The guaranteed solution is the trivial solution in which every variable is equal to 0. If a homogeneous system has a nontrivial solution, then it must meet a particular requirement involving the number of equations and number of variables in the system.
If a homogeneous system of linear equations has fewer equations than it has unknowns, then it has a nontrivial solution. Further, a homogeneous system of linear equations has a nontrivial solution if and only if the system has at least one free variable.
The other variables are then related to the free variable through some algebraic rules. Trivializing the situation with a trivial system The homogeneous system of equations shown next has only a trivial solution — no nontrivial solutions. When the variables are equal to 0, the solution is considered trivial.
Taking the trivial and adding some more solutions The next system of equations has nontrivial solutions. Even though, at first glance, you see three equations and three unknowns and each equation is set equal to 0, the system does meet the requirement that there be fewer equa- tions than unknowns.
The requirement is met because one of the equations was actually created by adding a multiple of one of the other equations to the third equation. In the next section, I show you how to determine when one equation is a linear combination of two others — resulting in one less equa- tion than in the original system.
So, back to a system of equations: The variable x3 is a free variable. How did I know this? Just trust me for now — I just want to show you how the nontrivial solutions work. I show you how to find these values in the next section. Choose any new value for k, and you get a new set of numbers. Formulating the form for a solution Now I get down to business and show you how to make the determinations regarding trivial and nontrivial solutions.
You can tell whether a system of homogeneous equations has only a trivial solution or if it indeed has non- trivial solutions. You can accomplish the task by observation in the case of small, simple systems or by changing the system to an echelon form for more complicated systems. Traveling the road to the trivial The following system of linear equations has only a trivial solution. Substituting that 0 back into the second original equation, you get that x2 also is equal to 0. And back-substituting into the first or third equation gives you 0 for x1, too.
The clue to the trivial solution in this case was the fact that the last row of the matrix had all but one 0. When you have all 0s in a row, you usually have a nontrivial solution if the elimina- tion of the row creates fewer equations than variables. Joining in the journey to the nontrivial When a homogeneous system of linear equations has nontrivial solutions, you usually have an infinite number of choices of numbers to satisfy the system.
You make a choice for the first number, and then the others fall in line behind that choice. You find the secondary numbers using rules involving algebraic expressions.
For example, consider the next system of four linear equations with four unknowns. First, the system of equations: And the corresponding augmented matrix: Now, performing row operations, I change the matrix to reduced row echelon form. Need a refresher on the row operations notation? Turn to Chapter 3.
Notice that, in the third step, the last row changes to all 0s, meaning that it was a linear combination of the other equations. The number of equations reduces to three, so nontrivial solutions are to be found.
Delving Into Linear Independence Independence means different things to different people. Every year, the United States celebrates Independence Day. The word independence, in math-speak, often has to do with a set of vectors and the relationship between the vectors in that set.
Chapter 7: Homing In on Homogeneous Systems and Linear Independence A collection of vectors is either linearly independent or linearly dependent. The vectors are linearly dependent if the equation has a solution when at least one of the scalars is not equal to 0.
The description of linear independence is another way of talking about homogeneous systems of linear equations. Instead of discussing the algebraic equations and the corresponding augmented matrix, the discus- sion now focuses on vectors and vector equations. True, you still use an augmented matrix in your investigations, but the matrix is now created from the vectors. Testing for dependence or independence For example, consider the following set of vectors and test whether the vec- tors in the set are linearly independent or linearly dependent.
Then write an augmented matrix with the vectors as columns, and perform row reductions. The set of vectors is linearly dependent. But you have many instances in which a linear combination of the vectors equals 0 when the vectors themselves are not 0.
I demonstrate the fact that you have more than just the trivial solution by going to the reduced form of the matrix, rewriting the vector equation.
The nontrivial solutions occur when you have fewer equations than variables. So, starting with just three vectors, I have more equations than variables. Characterizing linearly independent vector sets You always have the tried-and-true method of examining the relationships of vectors in a set using an augmented matrix. When in doubt, go to the matrix form. When applying the following guidelines, you always assume that the dimension of each vector in the set is the same.
Chapter 7: Homing In on Homogeneous Systems and Linear Independence Wondering when a set has only one vector A set, in mathematics, is a collection of objects.
A set can have any number of elements or objects, so a set may also contain just one element. And, yes, you can classify a set with one element as independent or dependent. A set containing only one vector is linearly independent if that one vector is not the zero vector. So, of the four sets shown next, just set D is dependent; the rest are all independent. Doubling your pleasure with two vectors A set containing two vectors may be linearly independent or dependent.
A set containing two vectors is linearly independent as long as one of the vec- tors is not a multiple of the other vector. Here are two sets containing two vectors. Set E is linearly independent. Set F is linearly dependent, because each element in the second vector is half that of the corresponding element in the first vector. And that dimension comes into play when making a quick observation about linear dependence or independence.
After making up this really awful example of vectors — just arbitrarily putting the number 1 and a bunch of prime numbers in for the elements — I began to wonder if I could demonstrate to you that the set really is linearly depen- dent, that one of the vectors is a linear combination of the others.
So, without showing you all the gory details feel free, of course, to check my work , the reduced echelon form of the augmented matrix I used is The last vector is a linear combination of the first four vectors.
Using the values in the last column of the echelon form as multipliers, you see the linear combination that creates the final vector. Zeroing in on linear dependence Having the zero vector as the only vector in a set is clearly grounds for your having linear dependence, but what about introducing the zero vector into an otherwise perfectly nice set of nonzero vectors?
Reducing the number of vectors in a set If you already have a linearly independent set of vectors, what happens if you remove one of the vectors from the set? Is the new, reduced set also linearly independent, or have you upset the plan? If vectors are linearly independent, then removing an arbitrary vector, vi , does not affect the linear independence.
Connecting Everything to Basis In Chapter 13, you find material on a mathematical structure called a vector space. In this chapter, I just deal with the vectors that belong in a vector space.
In Chapter 13, you find the other important processes and properties needed to establish a vector space. When you have a set of linearly inde- pendent vectors, you sort of have a core group of vectors from which other vectors are derived using linear combinations. But, when looking at a set of vectors, which are the core vectors and which are the ones created from the core? Is there a rhyme or reason?
The answers to these questions have to do with basis. You find information on linear independence earlier in this chapter. You find a complete discussion of the span of a set of vectors in Chapter 5.
But, to put span in just a few words for now: A vector v is in the span of a set of vec- tors, S, if v is the result of a linear combination of the vectors in S.
So the con- cept basis puts together two other properties or concepts: span and linear independence. Broadening your horizons with a natural basis The broadest example of a basis for a set involves unit vectors. The unit vectors are linearly independent, and you can find a linear combi- nation made up of the unit vectors to create any vector in R3. When the unit vectors are used as a basis for a vector space, you refer to this as the natural basis or standard basis.
In fact, I have a formula for determining which scalar multipliers in the linear combination correspond to a random vector. If you have some vector use the linear combination to create the vector where So, to create the vector representing the landing on the moon by the Apollo 11 crew, use the setup for the linear combination, substitute the target num- bers into the formulas for the scalars, and check by multiplying and adding.
Charting out the course for determining a basis A set of vectors B is a basis for another set of vectors V if the vectors in B have linear independence and if the vectors in B span V. Determining basis by spanning out in a search for span A set of vectors may be rather diminutive or immensely large. I show you some more manageable vector sets, for starters, and expand my horizons to the infinitely large.
Chapter 7: Homing In on Homogeneous Systems and Linear Independence For example, consider the eight vectors shown here and how I find a basis the vectors. The vectors are clearly not linearly independent, because you see more vec- tors than there are rows.
The set of unit vectors is a natural, but can I get away with fewer than three vectors for the particular basis? I have a procedure to determine just what might constitute a basis for a span.
If you have vectors v1, v2,. Construct the corresponding augmented matrix. Transform the matrix to reduced row echelon form. Identify the vectors in the original matrix corresponding to columns in the reduced matrix that contain leading 1s the first nonzero ele- ment in the row is a 1. So, using the eight vectors as a demonstration, I write the augmented matrix and perform row reductions. The first two columns correspond to the first two vectors in the vector set.
So a basis for the eight vectors is a set containing the first two vectors: But, wait a minute! What if you had written the vectors in a different order? What does that do to the basis? Consider the same vectors in a different order. Now I write the augmented matrix and perform row operations.
Chapter 7: Homing In on Homogeneous Systems and Linear Independence Again, the first two columns of the reduced matrix have leading 1s. So the first two vectors in the new listing can also be a basis for the span of vectors. Instead, write an expression in which the rule shown in the vector is the sum of two scalar multiples.
The two vectors in the linear combination are linearly independent, because neither is a multiple of the other. The vectors with the specifications can all be written as linear combinations of the two vectors. The basis must span the set — you must be able to construct linear combinations that result in all the vectors — and the vectors in the basis must be linearly independent. In this section, I explain how basis applies to matrices, in general, and even to polynomials.
Likewise, matri- ces in a basis must all have the same dimension. To determine if the four matrices form a basis for all the matrices in the span, you need to know if the matrices are linearly independent.
So the matrices are linearly independent, and the set S is the basis of the set of all matrices fitting the prescribed format.
Now I introduce P2, P3, P4, and so on to represent second-degree polynomials, third- degree polynomials, fourth-degree polynomials, and so on, respectively. Chapter 7: Homing In on Homogeneous Systems and Linear Independence A polynomial is the sum of variables and their coefficients where the variables are raised to whole-number powers. So the elements in the set form a basis for P2. Now consider a certain basis for P3. Solving for the values of each ai , The set Q spans a third-degree polynomial when the multipliers assume the values determined by the coefficients and constants in the polynomials.
Also, ematician, living in the late s through the Legendre worked with polynomials, beginning early s. He made significant contributions with discoveries involving the roots of polyno- to several areas of mathematics, statistics, and mials and culminating with establishing struc- physics. Getting his feet wet with the trajecto- tures called Legendre polynomials, which are ries of cannonballs, Legendre then moved on to found in applications of mathematics to physics various challenges in mathematics.
When you and engineering. Bastille in and continuing during the Legendre made some good starts, produc- French Revolution — losing all his money. He ing mathematics that were later completed or continued to work in mathematics and stayed proven by others. But, on his own, he is credited clear of any political activism during the revolu- with developing the least squares method used tion, producing significant contributions.
And, finally, to check for linear independence, determine if you have more than just the trivial solution. Solve the system of equations involving the mul- tipliers in which the linear combination is set equal to 0. The multipliers a1 and a2 are equal to 0. Only the trivial solution exists, so the elements are linearly independent, and Q is a basis for P3. Finding the dimension based on basis The dimension of a matrix or vector is tied to the number of rows and columns in that matrix or vector.
The dimension of a vector space see Chapter 13 , is the number of vectors in the basis of the vector space.
I discuss dimension here, to tie the concept to the overall picture of basis in this chapter. Look at the following two sets of vectors, F and T: So if set F is a basis for R4 and T is a basis for R2, then the span of F has dimension 4, because it contains four vectors, and T has dimension 2, because this basis contains two vectors.
You want to find the basis for V, which will help in determining more about set A. To find the basis, write the vectors in V as an augmented matrix, and go through row reductions. In reduced echelon form, the matrix has leading 1s in the first and third col- umns, corresponding to the first and third vectors in V. So the basis is and the dimension is 2. Perhaps you want to be able to extend your set of vectors to more than five.
Write the linear combinations of the vectors in the basis and the corresponding equations. You have a basis for R3. Little did you know if you were a Transformers fan that you were being set up for a fairly serious mathematical subject. Some transformations found in geometry are performed by moving objects around systematically and not changing their shape or size. Other transformations make more dramatic changes to geomet- ric figures. Linear transformations even incorporate some of the geometric transformational processes.
But linear transformations in this chapter have some restrictions, as well as opening up many more mathematical possibilities. In this chapter, I describe what linear transformations are. Then I take you through some examples of linear transformations. I show you many of the operational properties that accompany linear transformations.
And, finally, I describe the kernel and range of a linear transformation — two concepts that are very different but tied together by the transformation operator.
Formulating Linear Transformations Linear transformations are very specific types of processes in mathematics. They often involve mathematical structures such as vectors and matrices; the transformations also incorporate mathematical operations. Some of the operations used by linear transformations are your everyday addition and multiplication; others are specific to the type of mathematical structure that the operation is being performed upon.
In later sections, I show you examples of different types of linear transformations — even drawing pictures where appropriate. Delineating linear transformation lingo In mathematics, a transformation is an operation, function, or mapping in which one set of elements is transformed into another set of elements.
And the trigonomet- ric functions are truly amazing. The trig functions, such as sin x, take angle measures in degrees or radians and transform them into real numbers. A linear transformation is a particular type of transformation in which one set of vectors is transformed into another vector set using some linear operator.
Consider the following linear transformation, which I choose to name with the capital letter T. So, if you want T v when v is the following vector, you get The linear operator the rule describing the transformation of a linear trans- formation might also involve the multiplication of a matrix times the vector being operated upon.
The need for the correct dimension is covered fully in Chapter 3. For example, consider the transformation involving the matrix A, shown next.
Again, for how and why this change in dimension occurs after matrix multipli- cation, refer to the material in Chapter 3. Completing the picture with linear transformation requirements The two properties required to make a transformation perform as a linear transformation involve vector addition and scalar multiplication.
Both proper- ties require that you get the same result when performing the transformation on a sum or product after the operation as you do if you perform the transfor- mation and then the operation. I demonstrate the first of the two requirements — the additive requirement — needed for transformation T to be a linear transformation, using random vec- tors u and v.
First, the two vectors are added and the transformation per- formed on the result. Then I show how to perform the transformation on the two original vectors and add the transformed results. Chapter 8: Making Changes with Linear Transformations The end results are the same, as they should be. Again, the results are the same. I start with a linear transformation, so, of course, the rules work. But how do you know that a transformation is really a linear transformation?
Instead of trying to come up with all possible vectors which is usually impractical, if not impossible , you apply the transformation rule to some general vector and determine if the rules hold.
Establishing a process for determining whether you have a linear transformation For example, using the same transformation T, as described in the previous section, and two general vectors u and v, I first prove that the transformation of a vector sum is the same as the sum of the transformations on the vector.
T and the vectors u and v: I now determine if the rule involving addition and the transformation holds for all candidate vectors. The sums and differences in the two end results are the same. Confirmation is achieved. For example, the transformation W, shown next, fails when the addition property is applied. Proposing Properties of Linear Transformations The definition of a linear transformation involves two operations and their properties.
Many other properties of algebra also apply to linear operations when one or more vectors or transformations are involved. Some algebraic properties that you find associated with linear transformations are those of commutativity, associativity, distribution, and working with 0. Summarizing the summing properties In algebra, the associative property of addition establishes that, when adding the three terms x, y, and z, you get the same result by adding the sum of x and y to z as you do if you add x to the sum of y and z.
The associative property of addition applies to linear transformations when you perform more than one transformation on a vector. The commutative property of addition applies to linear transformations when you perform more than one transformation on a vector. See Chapter 3 for more on dimension and the addition of matrices. Introducing transformation composition and some properties Transformation composition is actually more like an embedded operation. When you perform the composition transformation T1 followed by transfor- mation T2, you first perform transformation T2 on the vector v and then per- form transformation T1 on the result.
Being able to perform transformation composition is dependent upon the vectors and matrices being of the correct dimension. T1 is then performed on that result. And, in general, the composition of these two transformations is Associating with vectors and the associative property of composition The associative property has to do with the grouping of the transformations, not the order. But, with careful arrangements of transfor- mations and dimension, you do get to see the associative property in action when composing transformations.
The associative property states that when you perform transformation composition on the first of two transformations and then the result on a third, you get the same result as performing the com- position on the last two transformations and then performing the transforma- tion described by the first on the result of those second two. Because multiplication of vectors and matrices must follow the rules involving proper dimensions, this associative property only applies when it makes any sense.
Scaling down the process with scalar multiplication Performing scalar multiplication on matrices or vectors amounts to multi- plying each element in the matrix or vector by a constant number. Because transformations performed on vec- tors result in other vectors, the properties of scalar multiplication do hold in transformation multiplication. In fact, the scalar can be introduced at any one of three places in the multiplication. Chapter 8: Making Changes with Linear Transformations Performing identity checks with identity transformations Addition and multiplication of real numbers include different identity ele- ments for each operation.
The identity for addition is 0, and the identity for multiplication is 1. Factor fearlessly, conquer the quadratic formula, and solve linear equations. There's no doubt that algebra can be easy to some while extremely challenging to others. 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.
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