# HW5 - MA/PH 607 Homework Handout V A Let i be a...

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09/23/2011 Homework Handout V A. Let be a two-dimensional traditional vector space with standard basis . i T œ ß e f i j Using this basis for , do the following (unless otherwise indicated, view as both the i i “input space” and the “output space”). 1 1. Find the matrix for the “magnification by 2” operator given by . ` # v v œ # 2. Find the matrix for the projection onto the vector, pr i i ² ± i @ œ @ " # " j i Þ 3. Find the matrix for the vector projection onto , pr , where 1 . Then a v a i j ² a use the matrix to find pr when . ² a v v i j œ ± ² % 4. Find the matrix for the linear operator such that and _ _ _ i i j j i œ # ² % œ \$ ² % j . v v i j Then use the matrix to find when _ œ ± ² 5. Find the matrix for the rotation of every vector counterclockwise by a fixed angle e ) ß ) e ) Þ œ œ ± ² % Then use the matrix to find when and ) v v i j 1 ' 6. Find the matrix for the rotation of every vector clockwise by a fixed angle . ) 7. Find the matrix for the dot product with , . (In this case, a i j v a v œ W a the “target space” is the space of all real numbers . Use as the basis for .) e f " B. Let be a three-dimensional traditional vector space with standard basis . i T œ ß ß e f i j k Using this basis for , do the following (unless otherwise indicated, view as both the i i “input space” and the “output space”). 1. Find the matrix for the “magnification by 2” operator given by . ` # v v œ # 2. Find the matrix for the vector projection onto , pr , where . a v a i j k ² a œ # ² Then use the matrix to find pr when . ² a v v i 7 j k œ ± ² 3. Find the matrix for the projection onto the plane spanned by the and vectors, i j ² ± pr e f i j ß " # \$ " # @ œ @ i k i j j . 4. Find the matrix for the rotation of every vector about by a fixed angle . e ) k ß ) ß k (This means that vectors parallel to do not move, while the vectors in the plane k spanned by are rotated — as in a previous exercise by the angle in the e f i j ß ) direction the fingers of your right hand would curl if your thumb points in the direction of .) (Compare with the above two-dimensional “clockwise rotation” problem.) k Then use the matrix to find when and . e ) k ß ) v v i j k œ œ ± ² 1 ' % 1 Be intelligent. Where posssible, find the matrix for the operation using the “boxed formula” or its slightly simplified version for the case where (in the notes i j œ Elementary Linear Transform Theory

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HW5 - MA/PH 607 Homework Handout V A Let i be a...

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