# HG changeset patch # User Rob Beezer # Date 1354512771 28800 # Node ID d15b1302390ed31edabbe75c2976e6be5416451b # Parent babb852ae23e81b6758aacff0c1cfadf3a4f8809 New Exercise VR.M20 diff --git a/changes.txt b/changes.txt --- a/changes.txt +++ b/changes.txt @@ -11,6 +11,7 @@ New: Diagram AIVS, Section IVLT New: Discussion following Theorem OBNM New: Exercises MM.T35, OD.T10, OD.T20 +New: Exercise VR.M20 (Tyler Ueltschi) Edit: Explained min(m,n) in Exercise PD.T15 Edit: More on induction in Proof Technique I (Dan Messenger) Edit: Added multiplicities to Solution EE.C19 (Duncan Bennett) diff --git a/src/bookinfo.xml b/src/bookinfo.xml --- a/src/bookinfo.xml +++ b/src/bookinfo.xml @@ -180,6 +180,14 @@ zoli.web.elte.hu + +Tyler +Ueltschi +U. of Puget Sound + + + + Andy Zimmer diff --git a/src/section-VR.xml b/src/section-VR.xml --- a/src/section-VR.xml +++ b/src/section-VR.xml @@ -847,6 +847,48 @@ + +The set \$B=\set{\vect{v}_1,\,\vect{v}_2,\,\vect{v}_3,\,\vect{v}_4}\$ is a basis of the vector space \$P_3\$, polynomials with degree 3 or less. Therefore \$\vectrepname{B}\$ is a linear transformation, according to . Find a formula for \$\vectrepname{B}\$. In other words, find an expression for \$\vectrep{B}{a+bx+cx^2+dx^3}\$. + +\vect{v}_11 - 5x - 22x^2 + 3x^3 + +\vect{v}_2-2 + 11x + 49x^2 - 8x^3\\ +\vect{v}_3-1 + 7x + 33x^2 - 8x^3 + +\vect{v}_4-1 + 4x + 16x^2 + x^3 + + +Our strategy is to determine the values of the linear transformation on a nice basis for the domain, and then apply the ideas of to obtain our formula. \$\vectrepname{B}\$ is a linear transformation of the form \$\ltdefn{\vectrepname{B}}{P_3}{\complex{4}}\$, so for a basis of the domain we choose a very simple one: \$C=\set{1,\,x,\,x^2,\,x^3}\$. We now give the vector representations of the elements of \$C\$, which are obtained by solving the relevant systems of equations obtained from linear combinations of the elements of \$B\$. + + + + + + + + + + +This is enough information to determine the linear transformation uniquely, and in particular, to allow us to use to construct a formula. + +\vectrep{B}{a+bx+cx^2+dx^3} +a\vectrep{B}{1} + b\vectrep{B}{x} + c\vectrep{B}{x^2} + d\vectrep{B}{x^3}\\ + +a\colvector{20 \\ 7 \\ 1 \\ 4} + +b\colvector{17 \\ 14 \\ -8 \\ -3} + +c\colvector{-3 \\ -3 \\ 2 \\ 1} + +d\colvector{0 \\ -1 \\ 1 \\ 1}\\ + +\colvector{ +20a + 17b - 3c\\ +7a + 14b - 3c - d\\ +a - 8b + 2c + d\\ +4a - 3b + c + d +} + + + +