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Math 1554 - 1.7-1.9 Notes

by: awesomenotes

Math 1554 - 1.7-1.9 Notes MATH 1554 K3

Georgia Tech

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About this Document

These notes cover linear independence and linear transformations
Linear Algebra
Stavros Garoufalidis
Class Notes
Ga Tech, Georgia Tech, Linear Algebra, math 1554, math 1553, Linear Independence, Linear transformations, matrix transformations
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This 3 page Class Notes was uploaded by awesomenotes on Monday February 15, 2016. The Class Notes belongs to MATH 1554 K3 at Georgia Institute of Technology taught by Stavros Garoufalidis in Spring 2016. Since its upload, it has received 34 views. For similar materials see Linear Algebra in Mathematics (M) at Georgia Institute of Technology.


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Date Created: 02/15/16
1.7 Linear Independence n - An indexed set of vectors {v ,…,1 } ip R is said to be linearly independent if the vector equation ???? ???? + ???? ???? + ⋯+ ???? ???? = 0 has only the trivial solution. ▯ ▯ ▯ ▯ ▯ ▯ The set {v 1…,v p is said be linearly dependent if their exist weights c 1…,c p not all zero, such that ????▯ ▯+ ???? ▯ ▯ ⋯+ ???? ???? =▯ ▯ - ???? ???? + ???? ???? + ⋯+ ???? ???? = 0 à linear dependence relation amount v ,…,v , when ▯ ▯ ▯ ▯ ▯ ▯ 1 p the weights are not all zero Linear Independence of Matrix Columns - The columns of a matrix A are linearly independent if and only if the equation Ax=0 has only the trivial solution Sets of One or Two Vectors - A set containing only one vector is linearly independent if and only if v is not the zero vector - You can always decide by inspection when a set of vectors in linearly dependent. Row operations are unnecessary. Just check whether at least one of the vectors is a scalar times the other (only applies to sets of two vectors) - A set of 2 vectors is linearly dependent if at least one of the vectors is a multiple of the other. The set is linearly dependent if and only if neither of the vectors is a multiple of the other - Geometric terms – two vectors are linearly dependent if and only if they lie on the same line through the origin Sets of Two or More Vectors Theorem 7 An indexed set ???? = {???? ,????,???? } ????f two or more vectors are linearly dependent if and only if at least one of the vectors in S is a linear combination of the others. In fact, if S is linearly dependent and ???? ≠ ????, then some v (jith j > 1) is a linear combination of the preceding vectors ???? ,…???????? ????▯???? Theorem 8 If a set contains more vectors than there are entries in each vector, then the set n is linearly dependent. That is, any set {v1,…,p } in R is linearly dependent if p>n Warning: Theorem 8 doesn’t say anything about the case where the number of vectors does not exceed the number of entries in each vector Theorem 9 n If a set ???? = ???????? ,…,???? } in R contains the zero vector, then the set is linearly dependent 1.8 Introduction to Linear Transformations n m - Transformation (or function or mapping) T from in R or R – a rule that assigned to each vector x in R a vector T(x) in Rm n - Domain of T – the set R m - Codomain of T – the set R - Notation – T: R à R ; indicated that the domain of T is R and the codomain m is R - Image (of a vector x under a transformation T) – the vector T(x) assigned to x by T - Range of T – set of all mages T(x) Matrix Transformations - Denoted as ???? ⊢ ???????? 2 2 - Shear transformation: transformation T: R à R defined by T(x) = Ax Linear Transformations - A transformation (or mapping) T is linear if: 1. ???? ???? + ???? = ???? ???? + ????(????), for all u,v in the domain of T 2. ???? ???????? = ????????(????), for all scalars c and all u in the domain of T - every matric transformation is a linear transformation - linear transformations preserve the operations of vector addition and scalar multiplication - if T is a linear transformation then: - ???? 0 = 0 - ???? ???????? + ???????? = ???????? ???? + ????????(????) for all vectors u,v in domain of T and scalars c,d - contraction – a mapping ???? ⊢ ???????? for some scalar r, with 0 ≤ ???? ≤ 1 - dilation – a mapping ???? ⊢ ???????? for some sclar =, with 1 < r 1.9 The Matrix of a Linear Transformation n n - Every linear transformation from R to R is actually a matrix transformation x à Ax and important properties of T are intimately related to familiar properties of A Theorem 10 n m Let T: R à R be a linear transformation. Then there exists a unique matrix A n such that T(x) = Ax for all x in R . In fact, A is in xhe m n matrix whose jth n column is the vector T(ej where e js the jth columns of the identity matrix in R A=[ T(e )… T(e )] 1 n - A=[ T(e )1 T(e )n – standard matrix for the linear transformation T - Linear transformation focuses on a mapping property - Matrix transformation – describes how mapping is implemented n m m m - The mapping of T: R à R is said to be onto R if each b in R is the image of at least one x in R n m m - The mapping of T: R à R is said to be one-to-one if each b in R is the n image of at most one x in R Theorem 11 n m Let T: R à R be a linear transformation. Then T is one-to-one if and only if the equation T(x) = 0 has only the trivial solution Theorem 12 n m Let T: R à R be a linear transformation, and let A be the standard matrix for T. Then: n m m 1. T maps R onto R if and only if the columns of A span R 2. T is one-to-one if and only if the columns of A are linearly independent


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