Chapter 4. Drawing lines: conditionals and coordinates in PostScript
... One reason this is not quite a trivial problem is that we are certainly not able to draw the entire infinite line. There is essentially only one way to draw parts of a line in PostScript, and that is to use @moveto@ and @lineto@ to draw a segment of the line, given two points on that line. Therefore ...
... One reason this is not quite a trivial problem is that we are certainly not able to draw the entire infinite line. There is essentially only one way to draw parts of a line in PostScript, and that is to use @moveto@ and @lineto@ to draw a segment of the line, given two points on that line. Therefore ...
2.2 Addition and Subtraction of Matrices and
... (A + B)C = AC + BC. Theorem 2.2.17 states that matrix multiplication is associative and distributive (over addition). We now consider the question of commutativity of matrix multiplication. If A is an m × n matrix and B is an n × m matrix, we can form both of the products AB and BA, which are m × m ...
... (A + B)C = AC + BC. Theorem 2.2.17 states that matrix multiplication is associative and distributive (over addition). We now consider the question of commutativity of matrix multiplication. If A is an m × n matrix and B is an n × m matrix, we can form both of the products AB and BA, which are m × m ...
Systems of Linear Equations in Fields
... Any admissible row operation is represented by some invertible matrix. Conversely, any invertible matrix represents some admissible row operation. To perform an elementary row operation on a matrix, one need only multiply on the left by the elementary matrix that represents the given row operation. ...
... Any admissible row operation is represented by some invertible matrix. Conversely, any invertible matrix represents some admissible row operation. To perform an elementary row operation on a matrix, one need only multiply on the left by the elementary matrix that represents the given row operation. ...
Linear algebra
Linear algebra is the branch of mathematics concerning vector spaces and linear mappings between such spaces. It includes the study of lines, planes, and subspaces, but is also concerned with properties common to all vector spaces.The set of points with coordinates that satisfy a linear equation forms a hyperplane in an n-dimensional space. The conditions under which a set of n hyperplanes intersect in a single point is an important focus of study in linear algebra. Such an investigation is initially motivated by a system of linear equations containing several unknowns. Such equations are naturally represented using the formalism of matrices and vectors.Linear algebra is central to both pure and applied mathematics. For instance, abstract algebra arises by relaxing the axioms of a vector space, leading to a number of generalizations. Functional analysis studies the infinite-dimensional version of the theory of vector spaces. Combined with calculus, linear algebra facilitates the solution of linear systems of differential equations.Techniques from linear algebra are also used in analytic geometry, engineering, physics, natural sciences, computer science, computer animation, and the social sciences (particularly in economics). Because linear algebra is such a well-developed theory, nonlinear mathematical models are sometimes approximated by linear models.