# Commits

committed d15b130

New Exercise VR.M20

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• Parent commits babb852

# File changes.txt

` 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)`

# File src/bookinfo.xml

` <econtact>zoli.web.elte.hu</econtact>`
` </contributorinfo>`
` `
`+<contributorinfo codename="tylerueltschi">`
`+<firstname>Tyler</firstname>`
`+<lastname>Ueltschi</lastname>`
`+<location>U. of Puget Sound</location>`
`+<econtact></econtact>`
`+</contributorinfo>`
`+`
`+`
` <contributorinfo codename="andyzimmer">`
` <firstname>Andy</firstname>`
` <lastname>Zimmer</lastname>`

# File src/section-VR.xml

` </problem>`
` </exercise>`
` `
`+<exercise type="M" number="20" rough="vector representation as formula">`
`+<problem contributor="tylerueltschi">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 <acroref type="theorem" acro="VRLT" />.  Find a <q>formula</q> for \$\vectrepname{B}\$.  In other words, find an expression for \$\vectrep{B}{a+bx+cx^2+dx^3}\$.`
`+<alignmath>`
`+\vect{v}_1<![CDATA[&=]]>1 - 5x - 22x^2 + 3x^3`
`+<![CDATA[&]]>`
`+\vect{v}_2<![CDATA[&=]]>-2 + 11x + 49x^2 - 8x^3\\`
`+\vect{v}_3<![CDATA[&=]]>-1 + 7x + 33x^2 - 8x^3`
`+<![CDATA[&]]>`
`+\vect{v}_4<![CDATA[&=]]>-1 + 4x + 16x^2 + x^3`
`+</alignmath>`
`+</problem>`
`+<solution contributor="robertbeezer">Our strategy is to determine the values of the linear transformation on a <q>nice</q> basis for the domain, and then apply the ideas of  <acroref type="theorem" acro="LTDB" /> 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\$.`
`+<alignmath>`
`+<![CDATA[\vectrep{B}{1}  &=\colvector{20 \\ 7 \\ 1 \\ 4}]]>`
`+<![CDATA[&]]>`
`+<![CDATA[\vectrep{B}{x}  &=\colvector{17 \\ 14 \\ -8 \\ -3}]]>`
`+<![CDATA[&]]>`
`+<![CDATA[\vectrep{B}{x^2}&=\colvector{-3 \\ -3 \\ 2 \\ 1}]]>`
`+<![CDATA[&]]>`
`+<![CDATA[\vectrep{B}{x^3}&=\colvector{0 \\ -1 \\ 1 \\ 1}]]>`
`+<![CDATA[&]]>`
`+</alignmath>`
`+This is enough information to determine the linear transformation uniquely, and in particular, to allow us to use <acroref type="theorem" acro="LTLC" /> to construct a formula.`
`+<alignmath>`
`+\vectrep{B}{a+bx+cx^2+dx^3} <![CDATA[&=]]>`
`+a\vectrep{B}{1} + b\vectrep{B}{x} + c\vectrep{B}{x^2} + d\vectrep{B}{x^3}\\`
`+<![CDATA[&=]]>`
`+a\colvector{20 \\ 7 \\ 1 \\ 4} +`
`+b\colvector{17 \\ 14 \\ -8 \\ -3} +`
`+c\colvector{-3 \\ -3 \\ 2 \\ 1} +`
`+d\colvector{0 \\ -1 \\ 1 \\ 1}\\`
`+<![CDATA[&=]]>`
`+\colvector{`
`+20a + 17b - 3c\\`
`+7a + 14b - 3c - d\\`
`+a - 8b + 2c + d\\`
`+4a - 3b + c + d`
`+}`
`+</alignmath>`
`+</solution>`
`+</exercise>`
`+`
` </exercisesubsection>`
` `
` </section>`