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# NUMERICAL LINEAR ALGEBRA MATH 526

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This 3 page Class Notes was uploaded by Cassidy Grimes on Monday October 26, 2015. The Class Notes belongs to MATH 526 at University of South Carolina - Columbia taught by Staff in Fall. Since its upload, it has received 36 views. For similar materials see /class/229542/math-526-university-of-south-carolina-columbia in Mathematics (M) at University of South Carolina - Columbia.

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Date Created: 10/26/15

MATH 526 Review Chapter 1 Chapter Chapter Solving linear system do elementary operations to nd reduced row echelon form RREF and by observing the RREF you should be able to determine some properties of the solution such as in nitely many unique or no solution What if you have more equations than unknowns ie for an n by m matrix A n gt m geometric view of linear systems with 2 unknowns if unique solution intersection of two lines if parallel no solution if overlap in nitly many solutions Do the same consideration for 3 unknowns How to t a curve to data Linear combination of vectors can span a vector subspace When you get free variables in solution of system decompose the solution to linear combination of vectors with letter coe icients free variables and then vectors give the basis for your solution space Linear dependence and independence see de nition on page 96 set linear combination to be equal to 0 If the only solution is 0 the vectorss are independent if there exist non zero solutions they are dependent More than n vectors in R must be dependent 7 Vectors span a space77 means anything in the space can be written as linear combination of the vectors Theorem 4 on page 110 has the same meaning matrix vector product should be thought of as a linear combination of column vectors of the matrix see de nition on page 114 Columns of A are indep77 is equivalent to Am 0 has only solution x 0 Matrix matrix product see page 143 for rules and properties AB 7 BA lnverse of matrix A may be invertible if A is square A is invertible Q columns indep Q detA 7 0 Q Am 0 has only solution x 0 Q we can nd square matrix B such that AB BAI See Theorem 5 on page 165 for properties of inverse and other conclusions involving the inverse Chapter Chapter Kernel and range of transformation or matrix both in de nition and geometic point of view When you do bunch of transformations to one guy the order does matter which means when you do matrix matrix multiplication the order does matter Linear transformation xes the origin like for matrix A T0 A0 is always 0 De nition of linear transformation on page 233 De nition of subspace 1 0 is in it 2 closed under scalar multiplication 3 closed under vector addition How to prove a set W is a subspace just check the 3 points in de nition 121 up in R then Spanv1 UP is a subspace of R A is m by 71 matrix then solution set of Am 0 nullspace is a subspace of R columns span a subspace of R kernel and range of a transformation are both subspaces of R for some 71 De nition of basis of W 1 vectors in basis span W 2 vectors are independent De nition of dimension of W the number of vectors in any basis How to prove vectors are independent 1 set the linear combination of them to O 2 try to show the only solutions of the above equation are all zeros If a set of vectors span a subspace then some subset of them is a basis for the subspace lf W has dimension p then 1 more than p vectors are de nitely dependent 2 less than p vectors won t span W 3 in W p vectors are independent if and only if they span W Every subspace space has a basis Suppose you know the coordinate under one basis how to nd the coordinate under another basis523 Theorem 23 24 and Fundamental Lemma De nition of column space row space and null space of a m by 71 matrix A Nullspace and column space of a matrix A is exactly the kernel and range of the trans formation de ned by A Find kernel and range geometrically page 284 4 How to nd basis for nullspaceby solving Am 0 How to nd columnrow space basis 543 and 544 or you can use transpose Column space and row space of a matrix A have the same dimension rank dim column space dim row space if A is m x n dim nullspace A is n 7 r n rank Theorem 36 and 9 for square invertible matrix You should generalize everything you learn before in any space Chapter Chapter Just be careful and double check What if you have a upperlower triangular matrix What if you have two rowscolumns the same What if you have one rowcolumn is a multiple of another rowcolumn An 71 by 71 matrix A 1 exchange two rows give you negative determinant 2 mul tiply one row multiply the determinant 3 adding a multiple of one row to another determinant remains the same Because detA detAT the above properties are also true for columns 77lnvertible77 equals non zero determinant A B both square then detAB detAdetBdetAT dam Cramer s Rule Eigenvector can t be 0 eigenvalue can be 0 To nd all evalues need to set the characteristic polynomial detA 7 AUX 0 For n by 71 matrix one should get characteristic polynomial of order n Always have n evalues but some of them can be the same Consider evalue and evector geometrically is evalue of A ltgt A 7 I is singular ie has 0 determinant espace of is the nullspace of A 7 A Know how to nd geometric and algebraic multiplicty 1 geo mult alg mult Know how to solve for evalues and the corresponding evectors eigenbasis if you have n distinct evalues always can nd ebasiswhy if some evalues are repeated it dependsuse EvectorsA see if you have columns Diagonalize a matrix One can diagonalize A if and only one can nd eigenbasis for A And put ebasis as columns to get P evalues ofA in diagonal to get K then P lAP K How do we know P is invertible Similar matrices They have same evalues why the reason is the de nition of similar and Theorem 12 on page 385

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