Appendix A
Vectors and Matrices
This appendix provides a refresher in vector and matrix algebra to support the main body of the book. Introductions to vectors and matrices are followed by descriptions of special matrix types, matrix inversion, and vector and matrix calculus.
A.1 Introduction to Vectors
A vector is a single-dimensional array of single-valued parameters, known as scalars.Here scalars are represented as italic and vectors as bold lower case. The scalarcomponents of a vector are denoted by the corresponding italic symbol with a single numerical index and are normally represented together as a bracketed column. Thus,
(A.1)
where, in this case, the vector has n components or elements. Vectors may also be represented with an underline, , or an arrow, , while many authors do not limit them to lower case. Sometimes, it is convenient to represent a vector column on one line. Here, the notation is used. A vector is often used to represent a quantity that has both magnitude and direction; these vectors usually have three components. However, the components of a vector may also be unrelated, with different units. Both types of vector are used here.
Vectors are added and subtracted by adding and subtracting the components:
(A.2)
The corresponding components must have the same units. A vector is multiplied by a scalar simply by multiplying each component by that scalar:
(A.3)
where two vectors have the same length, a scalar may be obtained by summing the products of the corresponding components. This is known as the scalar product or dot product and is written as
(A.4)
Each component product, , must have the same units. Scalar products have the properties
(A.5)
where the ...
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... are, respectively, the adjoint and determinant of A. For an m × n matrix, these are given by:
(A.33)
where r is an arbitrary row and is the minor, the determinant of A excluding row r and column i. The solution proceeds iteratively, noting that the determinant of a 2 × 2 matrix is Alternatively, many numerical methods for matrix inversion are available.
A.5 Calculus
The derivative of a vector or matrix with respect to a scalar simply comprises the derivatives of the elements. Thus,
(A.34)
The derivative of a scalar with respect to a vector is written as the transposed vector of the partial derivatives with respect to each vector component. Thus,
(A.35)
Post-multiplying this by the vector b then produces a scalar with the same units as a. The derivative of one vector with respect to another is then a matrix of the form:
(A.36)
The actual “matrix” described in the movie The Matrix provides the best example of a matrix. The “matrix” in the movie is a virtual reality implemented and created by machines to use human beings as a power supply. Their minds are trapped and they are unaware of the reality that they are simply batteries for the machines. Matrices are situations or surrounding circumstances within which something else originates, develops, or is contained. For example, in the explanation of the matrix above, human beings are in the matrix and their minds are contained within that matrix. However, containment is not the only type of matrix. Many different matrices exist in our lives. The educational system at California State University, Northridge can be considered a matrix. When students attend the university they start as one person, and through education and social experiences a new person is developed. Dick’s novel and The Matrix contain in their stories many mat...
...th the movie and the book have multiple matrices related with each other exactly the way matrices are related in mathematics. There are different dimensions of matrices in mathematics, and society alike, starting from single family and ending with multidimensional matrices of politics, religion,
The Matrix is considered by many people to be a cyberpunk triumph. Declan McCullagh from wired.com writes: "When Neo/Reeves wakes up from his VR slumber and unplugs from The Matrix, he joins a ragtag band of rebels led by the charismatic Morpheus (Lawrence Fishburne). Their plan: To overthrow the artificial intelligences that have robbed humanity of reality" (McCullagh). Entertainment weekly also sees The Matrix as a movie about rebellion against oppression: "Neo is, of course, The One, the prophesied leader of the oppressed who will lead the people of Zion (an underground city populated by the last free humans) from bondage--but only if he can believe in himself and trust in the power of love" (Bernadin).
The Matrix series is much more than an action-packed sci-fi thriller. After one view of this film for the second and third time, we start to notice a great deal of symbolism. This symbolism starts to paint a completely different picture than the images of humans battling machines. It is a religious story, with symbols deeply set in the Christian faith. The Matrix contains religious symbolism through its four main characters, Morpheus, Neo, Trinity and Cypher. In that each character personifies the “Father,” the “Son,” “Satan,” and the “Holy Spirit” of the Christian beliefs only shown through the amazing performances of the actors. A critic by the name of Shawn Levy said "The Matrix slams you back in your chair, pops open your eyes and leaves your jaw hanging slack in amazement."(metacritic.com)
The Matrix is a 1999 action film, noted for its science fiction and special effects, about the life of an individual who has been chosen to discover the truth of the world he lived in and eventually save all humanity from the enslavement of their minds in the Matrix. The story begins with an average computer programmer, named Thomas Anderson, who begins to notice strange occurrences as he dabbles in deeper into the secretive life of computer hacking and illegal software encryption through the nickname ‘Neo’. He is tracked down by another hacker, Trinity, and warns him of the dangers that would occur if he chose to remain in his current life. After Thomas realizes that he was being hunted down by sinister agents, he agrees to follow the path
2 To be concrete about the difference between the matrix and the "real world," I will refer to one as the matrix and the other as the ideology of the "real." The quotes are necessary as the ideology of the "real" is still a fictional ideology. Furthermore, it must be remembered that Althusser saw ideology as inescapable and a necessary feature of society ("there is no practice except by and in an ideology") (Althusser 93). Therefore, referring to the world outside of the matrix as the real world is insufficient and inaccurate. The ideology of the "real" (as Morpheus says, "welcome to the real") serves to enforce the notion of Neo not as rejecting ideology in favor of reality, but rather moving from the ideology of the machines (the matrix) to that of Morpheus (the ideology of the "real").
... matrix, in which color is a scalar multiple of the the vector objects contained within the array. As such, the syntax is determined by the mathematical properties of the entities in regards to how they arranged per their ascribed traits.
What is the matrix? The matrix is an artificial world, which has been pulled over to blind us from the truth, that we are slaves (Matrix,1999). We are trapped in a prison for our minds (Matrix,1999). We will never really get to feel, touch, or see anything for ourselves, except objects created through the matrix.
Wachowski, Andy, Dir. The Matrix. Perf. Reeves, Keanu, Lawrence Fishburne, and Carrie-Anne Moss. Warner Bros: 1999, Film.
A calculus is a calcified block attached to the tooth surface. It is based on the dental plaque on the surface of the tooth, which is formed by the gradual calcification of salt deposits in saliva.
The derivative of a function is the rate of change of that function. It shows how fast or how slow the function is changing. This can be useful in determining things such as instantaneous rates of change, velocity, acceleration and maximum profits. A good way to explain the concept of a derivative is to do it graphically. To illustrate, think of a drag car race. The track is only ¼ of a mile long, or 1320 feet. The dragster crosses the finish line in six seconds. How fast was the dragster going when it crossed the finish line? The dragster traveled 1320 feet in 6 seconds, so the average speed of the dragster is 1320 divided by 6 which equals 220 feet per second, or 150 miles per hour. The following graph represents the dragster’s position function as the red curve. The position function for the dragster is 36 2/3 x^2. The green line is the secant line connecting the dragster’s starting point and end point. The slope of this secant line is the average speed of the dragster, 220 feet per second, or 150 miles per hour.
This rule is derived from the power rule. The power rule is the most basic rule of differentiation. This rule is:
The Matrix is a narrative film by The Wachowski Brothers ,in 1999 is the groundbreaking visual effects film that tells the story of Neo, the hacker turned the One, in this a hero’s story. It is told in a chronological order from Neo’s point of view from the moment he wakes up to the moment he realizes that he is the One and the power it grants him. The film is best known for it use of bullet time and it commentary on perspective of our world and how real is it truly.
The Matrix is a sci-fi action film about a computer hacker named Neo that has been brought into another world deemed “the matrix.” The Matrix is a prime example of cinematography. The film uses many different types of cinematography such as mise-en-scene, special effects, and camera shots to make it interesting and entertaining to the audience guiding their attention to the important aspects of the film.
When the generated fields pass through magnetic materials which themselves contribute internal magnetic fields, ambiguities can arise about what part of the field comes from the external currents and what comes from the material itself. It is common to define another magnetic field quantity, usually called the "magnetic field strength" designated by H. It can be defined by the relationship, H = B0/μ0 = B/μ0 – M, and has the value of unambiguously designating the driving magnetic influence from external currents in a material, independent of the material's magnetic response. The relationship for B can be written in the equivalent form, B = μ0(H + M), H and M will have the same units, amperes/meter. To further distinguish B from H, B is sometimes called the magnetic flux density or the magnetic