Textbook Readings Unit One The Geometry of Linear Equations 1 1 2 1 Linear Combination cv dw Sometimes we want one particular combination Such as c 2 and d 1 that produces cv dw 4 5 Other times we want all the combinations of v and w The Vectors cv lie along a line When w is not on that line the combinations cv dw ll the whole two dimensional plane Column Vector v where v1 is the rst component and v2 is the second component Vector Addition v and w add to v w Scalar Multiplication 2v and v Combining addition with scalar multiplication we now form linear combinations of v and w Multiply v by c and multiply w by d then add cv dw De nition The sum of cv and dw is a linear combination of v and w Four special linear combinations 1v 1w sum of vectors 1v 1w difference of vectors 0v 0w zero vector The zero vector is always a possible combination Every time we have a space of vectors that zero vector will be included cv 0w vector cv in the direction of v Vectors in Three Dimensions The vector v corresponds to an arrow in three dimensional space Usually the arrow starts at the origin and ends at the point with the coordinates v1 v2 v3 All of the combinations of cu ll a line All the combinations of cu dv ll a plane All the combinations of cu dv ew ll a three dimensional space The dot product equals zero for perpendicular vectors De nition The length v of a vector v is the square root of the dot product De nition A unit vector u is a vector whose length equals one thus u u 1 Cosine Formula If v and w are nonzero vectors then Matrices Example Linear combinations The Matrix A acts on the vector x The result Ax is a combination b of the columns of A Dot products with rows Linear Equations The Inverse Matrix Ax b is solved by x A b Sb Independence and Dependence Independence w is not in the plane of u and v Dependence w is in the plane of u and v The important point is that the new vector w is a linear combination of u and v w u v By including w we get get no new vectors because w is already on the uv plane Review of the Key Ideas Matrix times a vector Ax is equal to the combination of all the columns of A The solutions to Ax b is x A b when A is an invertible matrix Elimination with Matrices 2 2 2 3 Elimination a systematic way to solve linear equations Before elimination x and y appear in both equations If w happens to be cu dv the third vector is in the plane of the rst two This means the combinations of u v w will not go outside that uv plane De nition The dot product of v v1 v2 and w w1 w2 is the number v w v1w1 v2w2 After elimination the rst unknown x has disappeared from the second equation Essential what we want is for the matrix to have only zeros below the diagonal To eliminate x Subtract a multiple of equation 1 from equation 2 Eliminations produces an upper triangular system this is the goal The system is solved from the bottom upwards using back substitution Pivot The rst equation contains 1x so the rst pivot the coef cient of x is 1 Multiplier The second equation contains 3x so the multiplier the coef cient of x is 3 You will see the multiplier rule if I change the rst equation to 4x 8x 4 Same straight line by the rst pivot becomes 4 The correct multiplier is now l 3 4 To nd the multiplier divide the coef cient 3 to be eliminated by the pivot 4 Pivot the rst nonzero in the row that does the elimination Multiplier entry to eliminate divided by pivot Breakdown of Elimination Zero is never allowed as a pivot There can be failure with in nitely many solutions or failure with no solutions Temporary failure zero in a pivot This calls for a row exchange Singular there is only one pivot Singular equations have no solution or in nitely many solutions Nonsingular there is a full set of pivots and exactly one solutions Three Equations in Three Unknowns Column 1 use the rst equation to create zeros below the rst pivot Column 2 use the second equation to create zeros below the second pivot Column 3 keep going to nd all n pivots and the U triangular Review of the Key Ideas A linear system Ax b becomes upper triangular Ux c after elimination We subtract l times equation j from equation i to make the i j entry zero The multiplier is l entry to eliminate row i pivot in row j Remember that pivots cannot be zero The zero is the pivot position can be xed if there is a nonzero below it The upper triangular system is solved by back substitution starting at the bottom When breakdown is permanent the system has no solution or in nitely many Elimination Using Matrices The goal is to express all the steps of elimination and the nal result in the clearest way possible The word entry for a matrix corresponds to component for a vector General rule a A i j is in the row i column j The elimination matrix E that subtracts a multiple c of row j from row i has the extra nonzero entry c in the i j position Row Exchange Matrix A row exchange is needed when zero is in the pivot position To exchange the rows we use a permutation matrix We can include b as an extra column and follow it through elimination The Augmented Matrix Review of the Key Ideas Special matrices Identity matrix I elimination E using l exchange matrix P Multiplying Ax b by E subtracts a multiple l of equation i from equation j The number l is the i j entry of the elimination matrix E For the augmented matrix A b that elimination step gives E A E b Matrix Operations and Inverses 2 4 2 5 To multiply AB If A has n columns B must have n rows The entry in the row i and column j of AB is row i of A column j of B Each column of AB is a combination of the columns of A The Laws for Matrix Operations Commutative law A B B A Distributive law c A B cA cB Associative law A B C A B C Block Multiplication Review of the Key Ideas The i j entry of AB is row i of A row j of B A times BC equals AB times C If the cuts between the columns of A match the cuts between the rows of B then block multiplication of AB is allowed Inverse Matrices The matrix A is invertible if there exists a matrix …
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