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Linear Algebra E3

What is a Linear Transformations? Let V and W be vector spaces. the mapping T:V --> W is called a LINEAR TRANSFORMATION if and only if: T(cu+v) = cT(u)+T(v) for every choice of u and v in V and scalars c in R.
When is T a linear operator? When V=W the T is a linear operator
What is the difference between an image and the range of a linear transformation? The image is of a vector. The range is the "map" (or collection of images from the transformation)
Define Image Fro each vector v in V, the vector w=T(v) is called the image of v under T
Define Range The range of an operator, T, denoted R(T), is the collection of all images of the vectors v in V R(T)={T(u)|v E V}
Define Null Space Let V and W be vector spaces, for a linear transformation T:V-->W, the NULL SPACE of T, denoted N(T), is defined as: N(T)={v E V|T(v)=0} "for every vector v in V the linear transformation of V maps v to the zero vector"
Null Space and Range Theorem 1. The null space of T is a subspace of V 2. The range of T is a subspace of W
Eigenvalue and Eigenvector Let A be an n x n matrix. A number \\ is called an EIGENVALUE of A provided that there exists a non zero vector in n space such that Av = \\v Every non zero vector satisfying this equation is called an eignevector of A corresponding to the eigenvalue \
Eigenvalue - Eigenvector Pair We say \\ and v form an eigenvalue - eigenvector pair: A will have infinitely many eigenvectors associated with \\ such that: A(cv)=c(Av)=c(\\v)=\\(cV)
Eigenspaces An eigenspace is the set of all eigenvectors corresponding to an eigenvalue along with the zero vector. V(\\)={v E n space | Av = \\v}
Define Dot Product Let u and v be vectors in euclidean n-space, the dot product of u and v is given by u*v = u1v1 + u2v2 + u3v3 +...+ unvn
Define length (norm) ||v|| = sqrt(v1^2 + v2^2 +...)
Define the distance between u and v ||u-v|| = sqrt((u-v)*(u-v))
Define unit vector u has a length = 1; thus ||u|| = 1 u = 1/||u|| * v
Define angle between u and v the cosine of the angle theta between the vectors v and w, is give by cos(theta) = (v*w)/(||v|| ||w||)
Define Orthogonal The vectors u and v are orthogonal if the angle between them is pi/2, cos(theta) = 0; thus (u*v)/(||u|| ||v||) = 0 and u*v = 0
Cauchy Schwartz Inequality |x*y| <= ||x|| ||y|| ie. the absolute value of two vectors is less than or equal to the product of the magnitude of 2 vectors
Define inner product space Let V be a vector space over all real numbers. An inner product on V is a function that associates with each pair of vectors in u and v in V a real number, denoted <u,v>, that satisfies all four inner product axioms
Inner Product Axioms 1. <u,u> => 0 and <u,u> = 0 if and only if u=0 (the zero vector) 2. <u,v> = <v,u> (symmetry) 3. <u+v,w> = <u,w> + <v,w> (linear property) 4. <cu,v> = c<u,v>
Inner Product Space A vector space V with an inner product
Orthogonal Sets We say that u and v are orthogonal provided that <u,v> = 0. The set V={v1,v2,v3,...vn} is orthogonal if the vectors are mutually orthogonal to each other
Define Mutually Orthogonal every vector in a set is orthogonal to each other
Created by: 1171208813