Vectors and MatricesIntroductionIntroductionIn addition to representing a vector in terms of its magnitude and direction, we can also represent a vector in terms of its components. This is illustrated in the figure at the right. Here we have a force vector, f, with a magnitude, |f|, and a direction, , relative to the x axis. (Note that the notation of the vector, f, and its magnitude, |f|, are different. The vector is the full specification of a magnitude and direction; e.g., 2000 pounds force at an angle of 30o from the x axis. The magnitude |f| is 2000 pounds in this example.) The components of the vector in the x and y directions are called fx and fy, respectively. These are not vectors, but are scalars that are multiplied by the unit vectors in the x and y direction to give the vector forces in the coordinate directions. The unit vectors in the x and y direction are usually given the symbols i and j, respectively. In this case we would write the vector in terms of its components as f = fxi + fyj. The vector components are called scalars to distinguish them from vectors. (Formally a scalar is defined as a quantity which is invariant under a coordinate transformation.)Basic matrix definitionsMatrix MultiplicationIntroduction to Simultaneous Linear Algebraic Equations// augment a matrix with b values// Upper triangular matrix complete; get x valuesMatrix Rank Determines Existence and Uniqueness of SolutionsDeterminantsInverse of a MatrixMatrix Eigenvalues and EigenvectorsMatrix Transformations Using EigenvectorsSimilarity transformationsIf pi is a constant, we can write this solution asIterative Solutions of Simultaneous Linear EquationsUsing equation [230] for ρJacobi gives the following resultSpecial matrices and quadratic formsVector spaces, norms, and inner productsCollege of Engineering and Computer ScienceMechanical Engineering DepartmentEngineering Analysis NotesLast updated: August 12, 2004 Larry CarettoVectors and MatricesIntroductionThese notes provide an introduction to the use of vectors and matrices in engineering analysis. In addition they provide a discussion of how the simple concept of a vector in mechanics leads to the concept of vector spaces for engineering analysis.Matrix notation is used to simplify the representation of linear algebraic equations. In addition, the matrix representation of systems of equations provides important properties regarding the system of equations. The discussion here presents many results without proof. You can refer to a general advanced engineering math text, like the one by Kreyszig or a text on linear algebra for such proofs.Parts of these notes have been prepared for use in a variety of courses to provide background information on the use of matrices in engineering problems. Consequently, some of the material may not be used in this course and different sections from these notes may be assigned at different times in the course.IntroductionA vector is a common concept in engineering mechanics that most students first saw in their high-school physics courses. Vectors are usually described in introductory courses as a quantity that has a magnitude and a direction. Force and velocity are common examples of vectors used in basic mechanics course.In addition to representing a vector in termsof its magnitude and direction, we can alsorepresent a vector in terms of itscomponents. This is illustrated in the figureat the right. Here we have a force vector, f,with a magnitude, |f|, and a direction, ,relative to the x axis. (Note that the notationof the vector, f, and its magnitude, |f|, aredifferent. The vector is the full specificationof a magnitude and direction; e.g., 2000pounds force at an angle of 30o from the xaxis. The magnitude |f| is 2000 pounds inthis example.) The components of the vector in the x and y directions are called fx and fy, respectively. These are not vectors, but are scalars that are multiplied by the unit vectors in the x and y direction to give the vector forces in the coordinate directions. The unit vectors in the x andy direction are usually given the symbols i and j, respectively. In this case we would write the vector in terms of its components as f = fxi + fyj. The vector components are called scalars to distinguish them from vectors. (Formally a scalar is defined as a quantity which is invariant undera coordinate transformation.)The concept of writing a vector in terms of its components is an important one in engineering analysis. Instead of writing f = fxi + fyj, we can write f = [fx fy], with the understanding that the first number is the x component of the vector and the second number is the y component of the vector. Using this notation we can write the unit vectors in the x and y directions as i = [1 0] and jEngineering Building Room 1333 Mail Code Phone: 818.677.6448E-mail: [email protected] 8348 Fax: 818.677.7062fyjfxifxyVectors and matrices L. S. Caretto, August 12, 2004 Page 2= [0 1]. This notation for unit vectors provides a link between representing a vector as a row or column matrix, as we will do below, and the conventional vector notation: f = fxi + fyj and f = [fx fy]. If we substitute i = [1 0] and j = [0 1] in the equation f = fxi + fyj, we get the result that f = fx[1, 0] +fy[0, 1] = [fx fy]. In place of the notation fx and fy for the x and y components, we can use numericalsubscripts for the coordinate directions and components. In this scheme we would call the x and y coordinate directions the x1 and x2 directions and the vector components would be labeled as f1 and f2. The numerical notation allows a generalization to systems with an arbitrary number of dimensions.From the diagram of the vector, f, and its components, we see that the magnitude of the vector, |f|, is given by Pythagoras’s theorem: 222122ffffyxf. We know that we can extend the two dimensional vector shown on the previous page to three dimensions. In this case our vectorshave three components, one in each coordinate direction. We can write the unit vectors in the three coordinate directions as i = [1 0 0], j = [0 1 0], and k = [0 0 1]. We would then write our three-dimensional vector, using numerical subscripts in place of x, y, and z subscripts, as f = f1i + f2j + f3k or f = [f1 fy f3]. If we substitute i = [1 0 0], j = [0 1 0], and k = [0, 0, 1] in the equation f = f1i + f2j + f3k, we get the result that f = f1[1 0 0] + f2[0 1 0] + f3[0 0