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## Matrices

by: Elaina Osinski

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# Matrices MATH 220

Elaina Osinski
Penn State
GPA 3.88

Staff

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## Popular in Mathematics (M)

This 0 page Class Notes was uploaded by Elaina Osinski on Sunday November 1, 2015. The Class Notes belongs to MATH 220 at Pennsylvania State University taught by Staff in Fall. Since its upload, it has received 9 views. For similar materials see /class/233007/math-220-pennsylvania-state-university in Mathematics (M) at Pennsylvania State University.

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Date Created: 11/01/15
Linear Combination and Span Section 44 0 De nitions from 44 1 Let 1117 1127 7 1 be vectors in a vector space V7 then any vector in the form 11111 12112 any where 117 127 7 an are scalars is called a linear combination of1117 1127 7 1 2 The vectors 1117 1127 7 1 in a vector space V7 are said to span V7 if every vector in V can be written as a linear combination of them That is7 for every 1 6 V7 there are scalars7 117 127 7 a such that va111a2112an11n 3 Let 1117 1127 7 11k be vectors in a vector space V7 the span of117 1127 7 11k is the set of all linear combinations of1117 1127 7 11k That is7 span117 1127 7 11k 1 1 11111 12112 unik where 117 127 7 an are arbitrary scalars 0 Theorem from 44 1 For any collection 1117 1127 7 wk of vectors in a vector space V7 the span of117 1127 7 11k is a subspace of V That is7 span 017 1127 7 11k is a subspace of V Linear Independence Section 45 0 De nitions from 45 1 Let 1117 1127 7 1 be vectors in a vector space V The vectors are said to be linearly dependent if there exists n scalars 017 027 7 077 not all zero such that 01111 0202 cnvn 0 If the vectors are not linearly dependent they are said to be linearly independent Another form of the above de nition Let 1117 1127 7 1 be vectors in a vector space V These vectors are said to be linearly independent if D 01111 02112 077177 0 1 implies that that 017 027 7 077 are all zero If equation 1 holds and one or more of the scalars is not zero7 then these vectors are said to be linearly dependent o Theorems from 45 H Let A be an n x 71 matrix with columns 1117 1127 7 1 The columns of A for a linearly independent set if and only if the only solution to Ax 0 is z 0 Let A be an n x 71 matrix Then detA 31 0 if and only if the columns and rows of A are linearly independent 3 Basis and Dimension Section 46 0 De nition from 46 1 D A nite set of vectors 017 1127 7 077 is a basis for a vector space V if a 017 1127 7 077 is linearly independent 117 127 The dimension of a vector space V is the number of vectors in every basis for V The dimension of a vector space is denoted dim V 7 en spans V If every basis has in nitely many vectors7 then V is said to be in nite dimensional7 and if the only vector in its basis is the zero vector then dim V 0 o Theorems from 46 1 D 00 117 127 lf117 1127 7 077 is a basis for a vector space V7 then any vector in V can be represented as a unique linear combination of the vectors in the basis 7 U77 and U17 u27 7 um are bases for V then m n This says that the number of terms in all bases for a vector space is the same If H is a subspace of a vector space V7 then dim H S dim V MATH 220 Similar Matrices and Diagonalization 0 De nition of Similar Matrices Two 71 by n matrices A and B are said to be similar if there exists an invertible n by 71 matrix C such that 304AO D The function de ned by equation 1 that takes the matrix A and returns the matrix B is called a similarity transformation Theorem 1 from 63 If A and B are similar 71 by n matrices then A and B have the same characteristic polynomial and therefore have the same eigenvalues 0 De nition of Diagonalizable An 71 by 71 matrix is diagonalizable if there is a diagonal matrix D such that A is similar to D 0 Theorem 2 from 63 An 71 by 71 matrix A is diagonalizable if and only if it has n linearly independent eigenvectors In that case the diagonal matrix D similar to A is A1 0 0 0 0 A2 0 0 D 7 0 0 A3 0 0 0 0 An where A1 A2 A are the eigenvalues of A Furthermore if C is a matrix whose columns are linearly independent eigenvectors of A then D04AO 4 2 0 Example For A lt 3 3 gt nd 0 which diagonalizes A by equation Verify that D C lAC 71 ab 71 d w note lt C d 77ad7b0ltic a If the n by 71 matrix A has n distinct eigenvalues then A is diagonalizable 0 Corollary from 63 0 Purpose of 64 Q What kind of matrices can be diagonalized A For starters real symmetric n by n matrices have n linearly independent eigenvectors and so they can be diagonalized

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