Review of Linear Independence Theorems
... the list linearly dependent (using the Linear Dependence Lemma). Proof: Suppose that L = (u1 , . . . , um ) is a linearly independent list in V . Since V is finite-dimensional, we can write V = span(w1 , . . . , wn ). Step 1. If w1 is in the span of L, throw it away. Otherwise, add w1 to the end of ...
... the list linearly dependent (using the Linear Dependence Lemma). Proof: Suppose that L = (u1 , . . . , um ) is a linearly independent list in V . Since V is finite-dimensional, we can write V = span(w1 , . . . , wn ). Step 1. If w1 is in the span of L, throw it away. Otherwise, add w1 to the end of ...
A I AI =
... Similarity is an equivalence relation that separates the set of all n-square matrices into equivalent classes. All matrices similar to a given matrix are similar to each other. What’s more? Any matrix similar to a given matrix represents the same linear transformation as the given matrix, but as ref ...
... Similarity is an equivalence relation that separates the set of all n-square matrices into equivalent classes. All matrices similar to a given matrix are similar to each other. What’s more? Any matrix similar to a given matrix represents the same linear transformation as the given matrix, but as ref ...
Mathematics 320
... This pleasant 50 minute test covers sections 12:1—6 of Calculus by Stewart (5ed). Relax and do well. Unless otherwise stated, problems are five points. Indicate your answers. ...
... This pleasant 50 minute test covers sections 12:1—6 of Calculus by Stewart (5ed). Relax and do well. Unless otherwise stated, problems are five points. Indicate your answers. ...
Applying transformations in succession Suppose that A and B are 2
... Corollary. The eigenvalues of A are the roots of the characteristic polynomial p. That is, the numbers λ such that p(λ) = 0. ...
... Corollary. The eigenvalues of A are the roots of the characteristic polynomial p. That is, the numbers λ such that p(λ) = 0. ...
Variable: a number that you don`t know, often represented by "x" or "y"
... Variable: a number that you don't know, often represented by "x" or "y" but any letter will do! Linear a mathematical statement that performs functions of addition, Expression: subtraction, multiplication, and division However, variable(s) in linear expressions ...
... Variable: a number that you don't know, often represented by "x" or "y" but any letter will do! Linear a mathematical statement that performs functions of addition, Expression: subtraction, multiplication, and division However, variable(s) in linear expressions ...
MISC-Math-Facts
... It will always be true that if the variable is on the left side of the inequality symbol, then shade the number line in the direction that the inequality symbol is pointing. However, if the variable is on the right side of the inequality symbol, then shade the number line in the opposite direction t ...
... It will always be true that if the variable is on the left side of the inequality symbol, then shade the number line in the direction that the inequality symbol is pointing. However, if the variable is on the right side of the inequality symbol, then shade the number line in the opposite direction t ...
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.