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{TEXT 204 25 "(a ) B has r nonzero rows," }{TEXT 204 0 "" }}{PARA 208 "" 0 "" {TEXT 204 18 "(b) B has columns " }{XPPEDIT 204 0 "Typesetting:-mrow(Typeset ting:-mi(\"\"), Typesetting:-msub(Typesetting:-mi(\"j\"), Typesetting: -mn(\"1\"), subscriptshift = \"0\", placeholder = \"false\"), Typesett ing:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF '6%-I#miGF$6#Q!F'-I%msubGF$6&-F,6#Q\"jF'-I#mnGF$6#Q\"1F'/%/subscriptsh iftGQ\"0F'/%,placeholderGQ&falseF'F+" }{TEXT 204 2 ", " }{XPPEDIT 204 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-msub(Typesett ing:-mi(\"j\"), Typesetting:-mn(\"2\"), subscriptshift = \"0\", placeh older = \"false\"), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG 6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-I%msubGF$6&-F,6#Q\"jF'-I #mnGF$6#Q\"2F'/%/subscriptshiftGQ\"0F'/%,placeholderGQ&falseF'F+" } {TEXT 204 7 ", ..., " }{XPPEDIT 204 0 "Typesetting:-mrow(Typesetting:- mi(\"\"), Typesetting:-msub(Typesetting:-mi(\"j\"), Typesetting:-mi(\" r\"), subscriptshift = \"0\", placeholder = \"false\"), Typesetting:-m i(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I# miGF$6#Q!F'-I%msubGF$6&-F,6#Q\"jF'-F,6#Q\"rF'/%/subscriptshiftGQ\"0F'/ %,placeholderGQ&falseF'F+" }{TEXT 204 57 " that are equal, respective ly, to the first r elements " }{XPPEDIT 204 0 "Typesetting:-mrow(Type setting:-mi(\"\"), Typesetting:-msub(Typesetting:-mi(\"e\"), Typesetti ng:-mn(\"1\"), subscriptshift = \"0\", placeholder = \"false\"), Types etting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_sysli bGF'6%-I#miGF$6#Q!F'-I%msubGF$6&-F,6#Q\"eF'-I#mnGF$6#Q\"1F'/%/subscrip tshiftGQ\"0F'/%,placeholderGQ&falseF'F+" }{TEXT 204 2 ", " }{XPPEDIT 204 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-msub(Type setting:-mi(\"e\"), Typesetting:-mn(\"2\"), subscriptshift = \"0\", pl aceholder = \"false\"), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulen ameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-I%msubGF$6&-F,6#Q\"e F'-I#mnGF$6#Q\"2F'/%/subscriptshiftGQ\"0F'/%,placeholderGQ&falseF'F+" }{TEXT 204 7 ", ..., " }{XPPEDIT 204 0 "Typesetting:-mrow(Typesetting: -mi(\"\"), Typesetting:-msub(Typesetting:-mi(\"e\"), Typesetting:-mi( \"r\"), subscriptshift = \"0\", placeholder = \"false\"), Typesetting: -mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%- I#miGF$6#Q!F'-I%msubGF$6&-F,6#Q\"eF'-F,6#Q\"rF'/%/subscriptshiftGQ\"0F '/%,placeholderGQ&falseF'F+" }{TEXT 204 27 " of the standard basis for " }{XPPEDIT 204 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetti ng:-msup(Typesetting:-mi(\"R\"), Typesetting:-mi(\"m\"), superscriptsh ift = \"0\"), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,T ypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-I%msupGF$6%-F,6#Q\"RF'-F,6#Q\" mF'/%1superscriptshiftGQ\"0F'F+" }{TEXT 204 1 "," }{TEXT 204 0 "" }} {PARA 208 "" 0 "" {TEXT 204 12 "(c) columns " }{XPPEDIT 2 0 "Typesetti ng:-mrow(Typesetting:-mi(\"\"), Typesetting:-msub(Typesetting:-mi(\"j \"), Typesetting:-mn(\"1\"), subscriptshift = \"0\", placeholder = \"f alse\"), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,Typese ttingGI(_syslibGF'6%-I#miGF$6#Q!F'-I%msubGF$6&-F,6#Q\"jF'-I#mnGF$6#Q\" 1F'/%/subscriptshiftGQ\"0F'/%,placeholderGQ&falseF'F+" }{TEXT 204 2 ", " }{XPPEDIT 2 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting :-msub(Typesetting:-mi(\"j\"), Typesetting:-mn(\"2\"), subscriptshift \+ = \"0\", placeholder = \"false\"), Typesetting:-mi(\"\"));" "-I%mrowG6 #/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-I%msubGF$ 6&-F,6#Q\"jF'-I#mnGF$6#Q\"2F'/%/subscriptshiftGQ\"0F'/%,placeholderGQ& falseF'F+" }{TEXT 204 7 ", ..., " }{XPPEDIT 2 0 "Typesetting:-mrow(Typ esetting:-mi(\"\"), Typesetting:-msub(Typesetting:-mi(\"j\"), Typesett ing:-mi(\"r\"), subscriptshift = \"0\", placeholder = \"false\"), Type setting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_sysl ibGF'6%-I#miGF$6#Q!F'-I%msubGF$6&-F,6#Q\"jF'-F,6#Q\"rF'/%/subscriptshi ftGQ\"0F'/%,placeholderGQ&falseF'F+" }{TEXT 204 35 " of A are linearly independent, and" }}{PARA 208 "" 0 "" {TEXT 204 24 "(d) if column k o f B is " }{XPPEDIT 204 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typ esetting:-msub(Typesetting:-mi(\"d\"), Typesetting:-mn(\"1\"), subscri ptshift = \"0\", placeholder = \"false\"), Typesetting:-mi(\"\"));" "- I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-I %msubGF$6&-F,6#Q\"dF'-I#mnGF$6#Q\"1F'/%/subscriptshiftGQ\"0F'/%,placeh olderGQ&falseF'F+" }{TEXT 204 0 "" }{XPPEDIT 204 0 "Typesetting:-mrow( Typesetting:-mi(\"\"), Typesetting:-msub(Typesetting:-mi(\"e\"), Types etting:-mn(\"1\"), subscriptshift = \"0\", placeholder = \"false\"), T ypesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_s yslibGF'6%-I#miGF$6#Q!F'-I%msubGF$6&-F,6#Q\"eF'-I#mnGF$6#Q\"1F'/%/subs criptshiftGQ\"0F'/%,placeholderGQ&falseF'F+" }{TEXT 204 3 " + " } {XPPEDIT 204 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:- msub(Typesetting:-mi(\"d\"), Typesetting:-mn(\"2\"), subscriptshift = \+ \"0\", placeholder = \"false\"), Typesetting:-mi(\"\"));" "-I%mrowG6#/ I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-I%msubGF$6& -F,6#Q\"dF'-I#mnGF$6#Q\"2F'/%/subscriptshiftGQ\"0F'/%,placeholderGQ&fa lseF'F+" }{TEXT 204 0 "" }{XPPEDIT 204 0 "Typesetting:-mrow(Typesettin g:-mi(\"\"), Typesetting:-msub(Typesetting:-mi(\"e\"), Typesetting:-mn (\"2\"), subscriptshift = \"0\", placeholder = \"false\"), Typesetting :-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6% -I#miGF$6#Q!F'-I%msubGF$6&-F,6#Q\"eF'-I#mnGF$6#Q\"2F'/%/subscriptshift GQ\"0F'/%,placeholderGQ&falseF'F+" }{TEXT 204 9 " + ... + " }{XPPEDIT 204 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-msub(Type setting:-mi(\"d\"), Typesetting:-mi(\"r\"), subscriptshift = \"0\", pl aceholder = \"false\"), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulen ameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-I%msubGF$6&-F,6#Q\"d F'-F,6#Q\"rF'/%/subscriptshiftGQ\"0F'/%,placeholderGQ&falseF'F+" } {TEXT 204 0 "" }{XPPEDIT 204 0 "Typesetting:-mrow(Typesetting:-mi(\"\" ), Typesetting:-msub(Typesetting:-mi(\"e\"), Typesetting:-mi(\"r\"), s ubscriptshift = \"0\", placeholder = \"false\"), Typesetting:-mi(\"\") );" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6# Q!F'-I%msubGF$6&-F,6#Q\"eF'-F,6#Q\"rF'/%/subscriptshiftGQ\"0F'/%,place holderGQ&falseF'F+" }{TEXT 204 33 ", then column k of A is equal to " }{XPPEDIT 204 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting: -msub(Typesetting:-mi(\"d\"), Typesetting:-mn(\"1\"), subscriptshift = \"0\", placeholder = \"false\"), Typesetting:-mi(\"\"));" "-I%mrowG6# /I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-I%msubGF$6 &-F,6#Q\"dF'-I#mnGF$6#Q\"1F'/%/subscriptshiftGQ\"0F'/%,placeholderGQ&f alseF'F+" }{TEXT 204 0 "" }{XPPEDIT 204 0 "Typesetting:-mrow(Typesetti ng:-mi(\"\"), Typesetting:-msub(Typesetting:-mi(\"a\"), Typesetting:-m row(Typesetting:-mi(\"\"), Typesetting:-msub(Typesetting:-mi(\"j\"), T ypesetting:-mn(\"1\"), subscriptshift = \"0\", placeholder = \"false\" ), Typesetting:-mi(\"\")), subscriptshift = \"0\", placeholder = \"fal se\"), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,Typesett ingGI(_syslibGF'6%-I#miGF$6#Q!F'-I%msubGF$6&-F,6#Q\"aF'-F#6%F+-F06&-F, 6#Q\"jF'-I#mnGF$6#Q\"1F'/%/subscriptshiftGQ\"0F'/%,placeholderGQ&false F'F+F@FCF+" }{TEXT 204 3 " + " }{XPPEDIT 204 0 "Typesetting:-mrow(Type setting:-mi(\"\"), Typesetting:-msub(Typesetting:-mi(\"d\"), Typesetti ng:-mn(\"2\"), subscriptshift = \"0\", placeholder = \"false\"), Types etting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_sysli bGF'6%-I#miGF$6#Q!F'-I%msubGF$6&-F,6#Q\"dF'-I#mnGF$6#Q\"2F'/%/subscrip tshiftGQ\"0F'/%,placeholderGQ&falseF'F+" }{TEXT 204 0 "" }{XPPEDIT 204 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-msub(Typeset ting:-mi(\"a\"), Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting: -msub(Typesetting:-mi(\"j\"), Typesetting:-mn(\"2\"), subscriptshift = \"0\", placeholder = \"false\"), Typesetting:-mi(\"\")), subscriptshi ft = \"0\", placeholder = \"false\"), Typesetting:-mi(\"\"));" "-I%mro wG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-I%msub GF$6&-F,6#Q\"aF'-F#6%F+-F06&-F,6#Q\"jF'-I#mnGF$6#Q\"2F'/%/subscriptshi ftGQ\"0F'/%,placeholderGQ&falseF'F+F@FCF+" }{TEXT 204 9 " + ... + " } {XPPEDIT 204 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:- msub(Typesetting:-mi(\"d\"), Typesetting:-mi(\"r\"), subscriptshift = \+ \"0\", placeholder = \"false\"), Typesetting:-mi(\"\"));" "-I%mrowG6#/ I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-I%msubGF$6& -F,6#Q\"dF'-F,6#Q\"rF'/%/subscriptshiftGQ\"0F'/%,placeholderGQ&falseF' F+" }{TEXT 204 0 "" }{XPPEDIT 204 0 "Typesetting:-mrow(Typesetting:-mi (\"\"), Typesetting:-msub(Typesetting:-mi(\"a\"), Typesetting:-mrow(Ty pesetting:-mi(\"\"), Typesetting:-msub(Typesetting:-mi(\"j\"), Typeset ting:-mi(\"r\"), subscriptshift = \"0\", placeholder = \"false\"), Typ esetting:-mi(\"\")), subscriptshift = \"0\", placeholder = \"false\"), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI( _syslibGF'6%-I#miGF$6#Q!F'-I%msubGF$6&-F,6#Q\"aF'-F#6%F+-F06&-F,6#Q\"j F'-F,6#Q\"rF'/%/subscriptshiftGQ\"0F'/%,placeholderGQ&falseF'F+F?FBF+" }{TEXT 204 1 "." }}}{EXCHG {PARA 221 "" 0 "" {TEXT 220 0 "" }}} {EXCHG {PARA 208 "" 0 "" {TEXT 204 194 "Generate a random 5 x 7 matrix of rank 4. We first generate a random 4 x 7 matrix, which likely has \+ rank 4, and then generate an additional row that is a linear combinati on of the existing rows." }{TEXT 204 0 "" }}}{EXCHG {PARA 216 "> " 0 " " {MPLTEXT 1 221 45 "A := RandomMatrix(4,7,generator=rand(-5..5));" } {MPLTEXT 1 221 0 "" }{MPLTEXT 1 221 2 "\n" }{MPLTEXT 1 221 1 "R" } {MPLTEXT 1 221 7 "ank(A);" }{MPLTEXT 1 221 0 "" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 221 108 "AA := ;" } {MPLTEXT 1 221 0 "" }{MPLTEXT 1 221 2 "\n" }{MPLTEXT 1 221 1 "R" } {MPLTEXT 1 221 8 "ank(AA);" }{MPLTEXT 1 221 0 "" }}}{EXCHG {PARA 208 " " 0 "" {TEXT 204 36 "We find the reduced row echelon form" }{TEXT 204 0 "" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 221 31 "B := ReducedRowE chelonForm(AA);" }{MPLTEXT 1 221 0 "" }}}{EXCHG {PARA 208 "" 0 "" {TEXT 204 97 "Note that the first four columns of the reduced matrix a re the first four standard basis vectors " }{XPPEDIT 204 0 "Typesettin g:-mrow(Typesetting:-mi(\"\"), Typesetting:-msub(Typesetting:-mi(\"e\" ), Typesetting:-mn(\"1\"), subscriptshift = \"0\", placeholder = \"fal se\"), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,Typesett ingGI(_syslibGF'6%-I#miGF$6#Q!F'-I%msubGF$6&-F,6#Q\"eF'-I#mnGF$6#Q\"1F '/%/subscriptshiftGQ\"0F'/%,placeholderGQ&falseF'F+" }{TEXT 204 2 ", " }{XPPEDIT 204 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting :-msub(Typesetting:-mi(\"e\"), Typesetting:-mn(\"2\"), subscriptshift \+ = \"0\", placeholder = \"false\"), Typesetting:-mi(\"\"));" "-I%mrowG6 #/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-I%msubGF$ 6&-F,6#Q\"eF'-I#mnGF$6#Q\"2F'/%/subscriptshiftGQ\"0F'/%,placeholderGQ& falseF'F+" }{TEXT 204 2 ", " }{XPPEDIT 204 0 "Typesetting:-mrow(Typese tting:-mi(\"\"), Typesetting:-msub(Typesetting:-mi(\"e\"), Typesetting :-mn(\"3\"), subscriptshift = \"0\", placeholder = \"false\"), Typeset ting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibG F'6%-I#miGF$6#Q!F'-I%msubGF$6&-F,6#Q\"eF'-I#mnGF$6#Q\"3F'/%/subscripts hiftGQ\"0F'/%,placeholderGQ&falseF'F+" }{TEXT 204 2 ", " }{XPPEDIT 204 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-msub(Typeset ting:-mi(\"e\"), Typesetting:-mn(\"4\"), subscriptshift = \"0\", place holder = \"false\"), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulename G6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-I%msubGF$6&-F,6#Q\"eF'- I#mnGF$6#Q\"4F'/%/subscriptshiftGQ\"0F'/%,placeholderGQ&falseF'F+" } {TEXT 204 27 " of the standard basis for " }{XPPEDIT 204 0 "Typesettin g:-mrow(Typesetting:-mi(\"\"), Typesetting:-msup(Typesetting:-mi(\"R\" ), Typesetting:-mn(\"5\"), superscriptshift = \"0\"), Typesetting:-mi( \"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#mi GF$6#Q!F'-I%msupGF$6%-F,6#Q\"RF'-I#mnGF$6#Q\"5F'/%1superscriptshiftGQ \"0F'F+" }{TEXT 204 296 "; the number of these match the rank of A. Th us by (c) the first four columns of A are linearly independent. The fi fth through seventh columns of B are obvious linear combinations of th e first four columns - and by (d) the coefficients in these linear com binations also apply to the columns of A." }{TEXT 204 0 "" }}{PARA 208 "" 0 "" {TEXT 204 6 "Check:" }{TEXT 204 0 "" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 221 82 "B[1,5]*Column(A,1) + B[2,5]*Column(A,2) + B[3 ,5]*Column(A,3) + B[4,5]*Column(A,4);" }{MPLTEXT 1 221 0 "" }{MPLTEXT 1 221 2 "\n" }{MPLTEXT 1 221 12 "Column(A,5);" }{MPLTEXT 1 221 0 "" }} }{EXCHG {PARA 208 "" 0 "" {TEXT 204 174 "The 5th column of A *is* the \+ linear combination of the first four columns of A, with coefficients c oming from the entries of the 5th column of B. Check the other two col umns:" }{TEXT 204 0 "" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 221 82 "B[1,6]*Column(A,1) + B[2,6]*Column(A,2) + B[3,6]*Column(A,3) + B[4 ,6]*Column(A,4);" }{MPLTEXT 1 221 0 "" }{MPLTEXT 1 221 2 "\n" } {MPLTEXT 1 221 12 "Column(A,6);" }{MPLTEXT 1 221 0 "" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 221 82 "B[1,7]*Column(A,1) + B[2,7]*Column(A, 2) + B[3,7]*Column(A,3) + B[4,7]*Column(A,4);" }{MPLTEXT 1 221 0 "" } {MPLTEXT 1 221 2 "\n" }{MPLTEXT 1 221 12 "Column(A,7);" }{MPLTEXT 1 221 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 227 8 "Exercise" }}{EXCHG {PARA 0 "" 0 "" {TEXT 213 122 "1) Generate a random 6 x 5 matrix of r ank 4, find its reduced row echelon form, and check that (d) holds for your matrix." }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "" "%#%?G" }} }}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 221 0 "" }}}{EXCHG {PARA 208 " " 0 "" {TEXT 204 57 "Now apply the theorem to the following problem si tuation." }{TEXT 204 0 "" }}{PARA 208 "" 0 "" {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 217 29 "The following polynomials in " }{XPPEDIT 217 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-msub(Typeset ting:-mi(\"P\"), Typesetting:-mn(\"3\"), subscriptshift = \"0\", place holder = \"false\"), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulename G6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-I%msubGF$6&-F,6#Q\"PF'- I#mnGF$6#Q\"3F'/%/subscriptshiftGQ\"0F'/%,placeholderGQ&falseF'F+" } {TEXT 217 45 " all have derivative equal to zero at x = 1: " } {XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-m row(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mi(\"\"), Ty pesetting:-msup(Typesetting:-mi(\"x\"), Typesetting:-mn(\"2\"), supers criptshift = \"0\"), Typesetting:-mi(\"\")), Typesetting:-mo(\"− \", form = \"infix\", fence = \"false\", separator = \"false\", lspace = \"mediummathspace\", rspace = \"mediummathspace\", stretchy = \"fal se\", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", \+ largeop = \"false\", movablelimits = \"false\", accent = \"false\", fo nt_style_name = \"2D Comment\", size = \"12\", foreground = \"[0,0,0] \", background = \"[255,255,255]\"), Typesetting:-mrow(Typesetting:-mn (\"2\"), Typesetting:-mo(\"⁢\", form = \"infix\", fence = \"false\", separator = \"false\", lspace = \"0em\", rspace = \"0em \", stretchy = \"false\", symmetric = \"false\", maxsize = \"infinity \", minsize = \"1\", largeop = \"false\", movablelimits = \"false\", a ccent = \"false\", font_style_name = \"2D Comment\", size = \"12\", fo reground = \"[0,0,0]\", background = \"[255,255,255]\"), Typesetting:- mi(\"x\")), Typesetting:-mi(\"\")), Typesetting:-mo(\",\", form = \"in fix\", fence = \"false\", separator = \"true\", lspace = \"0em\", rspa ce = \"verythickmathspace\", stretchy = \"false\", symmetric = \"false \", maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", mova blelimits = \"false\", accent = \"false\", font_style_name = \"2D Comm ent\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,25 5,255]\"), Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-mrow( Typesetting:-mi(\"\"), Typesetting:-msup(Typesetting:-mi(\"x\"), Types etting:-mn(\"3\"), superscriptshift = \"0\"), Typesetting:-mi(\"\")), \+ Typesetting:-mo(\"−\", form = \"infix\", fence = \"false\", sepa rator = \"false\", lspace = \"mediummathspace\", rspace = \"mediummath space\", stretchy = \"false\", symmetric = \"false\", maxsize = \"infi nity\", minsize = \"1\", largeop = \"false\", movablelimits = \"false \", accent = \"false\", font_style_name = \"2D Comment\", size = \"12 \", foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Typeset ting:-mrow(Typesetting:-mn(\"3\"), Typesetting:-mo(\"⁢ \", form = \"infix\", fence = \"false\", separator = \"false\", lspace = \"0em\", rspace = \"0em\", stretchy = \"false\", symmetric = \"fals e\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", mov ablelimits = \"false\", accent = \"false\", font_style_name = \"2D Com ment\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,2 55,255]\"), Typesetting:-mi(\"x\")), Typesetting:-mo(\"+\", form = \"i nfix\", fence = \"false\", separator = \"false\", lspace = \"mediummat hspace\", rspace = \"mediummathspace\", stretchy = \"false\", symmetri c = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"fa lse\", movablelimits = \"false\", accent = \"false\", font_style_name \+ = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", background \+ = \"[255,255,255]\"), Typesetting:-mn(\"1\")), Typesetting:-mo(\",\", \+ form = \"infix\", fence = \"false\", separator = \"true\", lspace = \" 0em\", rspace = \"verythickmathspace\", stretchy = \"false\", symmetri c = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"fa lse\", movablelimits = \"false\", accent = \"false\", font_style_name \+ = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", background \+ = \"[255,255,255]\"), Typesetting:-mrow(Typesetting:-mi(\"\"), Typeset ting:-mrow(Typesetting:-mi(\"\"), Typesetting:-msup(Typesetting:-mi(\" x\"), Typesetting:-mn(\"3\"), superscriptshift = \"0\"), Typesetting:- mi(\"\")), Typesetting:-mo(\"−\", form = \"infix\", fence = \"fa lse\", separator = \"false\", lspace = \"mediummathspace\", rspace = \+ \"mediummathspace\", stretchy = \"false\", symmetric = \"false\", maxs ize = \"infinity\", minsize = \"1\", largeop = \"false\", movablelimit s = \"false\", accent = \"false\", font_style_name = \"2D Comment\", s ize = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255]\" ), Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-msup(Typesett ing:-mi(\"x\"), Typesetting:-mn(\"2\"), superscriptshift = \"0\"), Typ esetting:-mi(\"\")), Typesetting:-mo(\"−\", form = \"infix\", fe nce = \"false\", separator = \"false\", lspace = \"mediummathspace\", \+ rspace = \"mediummathspace\", stretchy = \"false\", symmetric = \"fals e\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", mov ablelimits = \"false\", accent = \"false\", font_style_name = \"2D Com ment\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,2 55,255]\"), Typesetting:-mi(\"x\")), Typesetting:-mo(\",\", form = \"i nfix\", fence = \"false\", separator = \"true\", lspace = \"0em\", rsp ace = \"verythickmathspace\", stretchy = \"false\", symmetric = \"fals e\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", mov ablelimits = \"false\", accent = \"false\", font_style_name = \"2D Com ment\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,2 55,255]\"), Typesetting:-mn(\"1\"), Typesetting:-mo(\",\", form = \"in fix\", fence = \"false\", separator = \"true\", lspace = \"0em\", rspa ce = \"verythickmathspace\", stretchy = \"false\", symmetric = \"false \", maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", mova blelimits = \"false\", accent = \"false\", font_style_name = \"2D Comm ent\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,25 5,255]\"), Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-mrow( Typesetting:-mi(\"\"), Typesetting:-msup(Typesetting:-mi(\"x\"), Types etting:-mn(\"3\"), superscriptshift = \"0\"), Typesetting:-mi(\"\")), \+ Typesetting:-mo(\"−\", form = \"infix\", fence = \"false\", sepa rator = \"false\", lspace = \"mediummathspace\", rspace = \"mediummath space\", stretchy = \"false\", symmetric = \"false\", maxsize = \"infi nity\", minsize = \"1\", largeop = \"false\", movablelimits = \"false \", accent = \"false\", font_style_name = \"2D Comment\", size = \"12 \", foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Typeset ting:-mrow(Typesetting:-mi(\"\"), Typesetting:-msup(Typesetting:-mi(\" x\"), Typesetting:-mn(\"2\"), superscriptshift = \"0\"), Typesetting:- mi(\"\")), Typesetting:-mo(\"−\", form = \"infix\", fence = \"fa lse\", separator = \"false\", lspace = \"mediummathspace\", rspace = \+ \"mediummathspace\", stretchy = \"false\", symmetric = \"false\", maxs ize = \"infinity\", minsize = \"1\", largeop = \"false\", movablelimit s = \"false\", accent = \"false\", font_style_name = \"2D Comment\", s ize = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255]\" ), Typesetting:-mi(\"x\"), Typesetting:-mo(\"−\", form = \"infix \", fence = \"false\", separator = \"false\", lspace = \"mediummathspa ce\", rspace = \"mediummathspace\", stretchy = \"false\", symmetric = \+ \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false \", movablelimits = \"false\", accent = \"false\", font_style_name = \+ \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", background = \+ \"[255,255,255]\"), Typesetting:-mn(\"1\")), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6--I#miGF$6#Q!F' -F#6'F+-F#6%F+-I%msupGF$6%-F,6#Q\"xF'-I#mnGF$6#Q\"2F'/%1superscriptshi ftGQ\"0F'F+-I#moGF$63Q(−F'/%%formGQ&infixF'/%&fenceGQ&falseF'/%* separatorGFI/%'lspaceGQ0mediummathspaceF'/%'rspaceGFN/%)stretchyGFI/%* symmetricGFI/%(maxsizeGQ)infinityF'/%(minsizeGQ\"1F'/%(largeopGFI/%.mo vablelimitsGFI/%'accentGFI/%0font_style_nameGQ+2D~CommentF'/%%sizeGQ#1 2F'/%+foregroundGQ([0,0,0]F'/%+backgroundGQ.[255,255,255]F'-F#6%F9-FA6 3Q1⁢F'FDFGFJ/FMQ$0emF'/FPF]pFQFSFUFXFenFgnFinF[oF^oFaoF doF6F+-FA63Q\",F'FDFG/FKQ%trueF'F\\p/FPQ3verythickmathspaceF'FQFSFUFXF enFgnFinF[oF^oFaoFdo-F#6(F+-F#6%F+-F46%F6-F:6#Q\"3F'F=F+F@-F#6%F\\qFio F6-FA63Q\"+F'FDFGFJFLFOFQFSFUFXFenFgnFinF[oF^oFaoFdo-F:6#FZF_p-F#6(F+F hpF@F1F@F6F_pFdqF_p-F#6*F+FhpF@F1F@F6F@FdqF+" }{TEXT 217 54 ". Find a \+ basis for the subspace of all polynomials in " }{XPPEDIT 217 0 "Typese tting:-mrow(Typesetting:-mi(\"\"), Typesetting:-msub(Typesetting:-mi( \"P\"), Typesetting:-mn(\"3\"), subscriptshift = \"0\", placeholder = \+ \"false\"), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,Typ esettingGI(_syslibGF'6%-I#miGF$6#Q!F'-I%msubGF$6&-F,6#Q\"PF'-I#mnGF$6# Q\"3F'/%/subscriptshiftGQ\"0F'/%,placeholderGQ&falseF'F+" }{TEXT 217 168 " having derivative zero at x = 1 that consists of polynomials in this list, and write the remaining polynomials as linear combinations of the basis vectors you choose." }{TEXT 217 0 "" }}{PARA 208 "" 0 "" {TEXT 204 0 "" }}{PARA 208 "" 0 "" {TEXT 204 75 "The first step is to set up to check linear independence, that is, to solve" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 217 0 "" }{XPPEDIT 217 0 "Typesetting:-mr ow(Typesetting:-mi(\"\"), Typesetting:-msub(Typesetting:-mi(\"a\"), Ty pesetting:-mn(\"1\"), subscriptshift = \"0\", placeholder = \"false\") , Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI (_syslibGF'6%-I#miGF$6#Q!F'-I%msubGF$6&-F,6#Q\"aF'-I#mnGF$6#Q\"1F'/%/s ubscriptshiftGQ\"0F'/%,placeholderGQ&falseF'F+" }{TEXT 217 1 "(" } {XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-m row(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mi(\"\"), Ty pesetting:-msup(Typesetting:-mi(\"x\"), Typesetting:-mn(\"2\"), supers criptshift = \"0\"), Typesetting:-mi(\"\")), Typesetting:-mo(\"− \", form = \"infix\", fence = \"false\", separator = \"false\", lspace = \"mediummathspace\", rspace = \"mediummathspace\", stretchy = \"fal se\", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", \+ largeop = \"false\", movablelimits = \"false\", accent = \"false\", fo nt_style_name = \"2D Comment\", size = \"12\", foreground = \"[0,0,0] \", background = \"[255,255,255]\"), Typesetting:-mrow(Typesetting:-mn (\"2\"), Typesetting:-mo(\"⁢\", form = \"infix\", fence = \"false\", separator = \"false\", lspace = \"0em\", rspace = \"0em \", stretchy = \"false\", symmetric = \"false\", maxsize = \"infinity \", minsize = \"1\", largeop = \"false\", movablelimits = \"false\", a ccent = \"false\", font_style_name = \"2D Comment\", size = \"12\", fo reground = \"[0,0,0]\", background = \"[255,255,255]\"), Typesetting:- mi(\"x\")), Typesetting:-mi(\"\")), Typesetting:-mi(\"\"));" "-I%mrowG 6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-F#6'F+-F #6%F+-I%msupGF$6%-F,6#Q\"xF'-I#mnGF$6#Q\"2F'/%1superscriptshiftGQ\"0F' F+-I#moGF$63Q(−F'/%%formGQ&infixF'/%&fenceGQ&falseF'/%*separator GFI/%'lspaceGQ0mediummathspaceF'/%'rspaceGFN/%)stretchyGFI/%*symmetric GFI/%(maxsizeGQ)infinityF'/%(minsizeGQ\"1F'/%(largeopGFI/%.movablelimi tsGFI/%'accentGFI/%0font_style_nameGQ+2D~CommentF'/%%sizeGQ#12F'/%+for egroundGQ([0,0,0]F'/%+backgroundGQ.[255,255,255]F'-F#6%F9-FA63Q1&Invis ibleTimes;F'FDFGFJ/FMQ$0emF'/FPF]pFQFSFUFXFenFgnFinF[oF^oFaoFdoF6F+F+" }{TEXT 217 4 ") + " }{XPPEDIT 217 0 "Typesetting:-mrow(Typesetting:-m i(\"\"), Typesetting:-msub(Typesetting:-mi(\"a\"), Typesetting:-mn(\"2 \"), subscriptshift = \"0\", placeholder = \"false\"), Typesetting:-mi (\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#m iGF$6#Q!F'-I%msubGF$6&-F,6#Q\"aF'-I#mnGF$6#Q\"2F'/%/subscriptshiftGQ\" 0F'/%,placeholderGQ&falseF'F+" }{TEXT 217 1 "(" }{XPPEDIT 18 0 "Typese tting:-mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mi( \"\"), Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-msup(Type setting:-mi(\"x\"), Typesetting:-mn(\"3\"), superscriptshift = \"0\"), Typesetting:-mi(\"\")), Typesetting:-mo(\"−\", form = \"infix\" , fence = \"false\", separator = \"false\", lspace = \"mediummathspace \", rspace = \"mediummathspace\", stretchy = \"false\", symmetric = \" false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", font_style_name = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", background = \"[2 55,255,255]\"), Typesetting:-mrow(Typesetting:-mn(\"3\"), Typesetting: -mo(\"⁢\", form = \"infix\", fence = \"false\", separat or = \"false\", lspace = \"0em\", rspace = \"0em\", stretchy = \"false \", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", la rgeop = \"false\", movablelimits = \"false\", accent = \"false\", font _style_name = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Typesetting:-mi(\"x\")), Typesetting :-mo(\"+\", form = \"infix\", fence = \"false\", separator = \"false\" , lspace = \"mediummathspace\", rspace = \"mediummathspace\", stretchy = \"false\", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", movablelimits = \"false\", accent = \"fal se\", font_style_name = \"2D Comment\", size = \"12\", foreground = \" [0,0,0]\", background = \"[255,255,255]\"), Typesetting:-mn(\"1\")), T ypesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_s yslibGF'6%-I#miGF$6#Q!F'-F#6(F+-F#6%F+-I%msupGF$6%-F,6#Q\"xF'-I#mnGF$6 #Q\"3F'/%1superscriptshiftGQ\"0F'F+-I#moGF$63Q(−F'/%%formGQ&infi xF'/%&fenceGQ&falseF'/%*separatorGFI/%'lspaceGQ0mediummathspaceF'/%'rs paceGFN/%)stretchyGFI/%*symmetricGFI/%(maxsizeGQ)infinityF'/%(minsizeG Q\"1F'/%(largeopGFI/%.movablelimitsGFI/%'accentGFI/%0font_style_nameGQ +2D~CommentF'/%%sizeGQ#12F'/%+foregroundGQ([0,0,0]F'/%+backgroundGQ.[2 55,255,255]F'-F#6%F9-FA63Q1⁢F'FDFGFJ/FMQ$0emF'/FPF]pFQF SFUFXFenFgnFinF[oF^oFaoFdoF6-FA63Q\"+F'FDFGFJFLFOFQFSFUFXFenFgnFinF[oF ^oFaoFdo-F:6#FZF+" }{TEXT 217 4 ") + " }{XPPEDIT 217 0 "Typesetting:-m row(Typesetting:-mi(\"\"), Typesetting:-msub(Typesetting:-mi(\"a\"), T ypesetting:-mn(\"3\"), subscriptshift = \"0\", placeholder = \"false\" ), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingG I(_syslibGF'6%-I#miGF$6#Q!F'-I%msubGF$6&-F,6#Q\"aF'-I#mnGF$6#Q\"3F'/%/ subscriptshiftGQ\"0F'/%,placeholderGQ&falseF'F+" }{TEXT 217 1 "(" } {XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-m row(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mi(\"\"), Ty pesetting:-msup(Typesetting:-mi(\"x\"), Typesetting:-mn(\"3\"), supers criptshift = \"0\"), Typesetting:-mi(\"\")), Typesetting:-mo(\"− \", form = \"infix\", fence = \"false\", separator = \"false\", lspace = \"mediummathspace\", rspace = \"mediummathspace\", stretchy = \"fal se\", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", \+ largeop = \"false\", movablelimits = \"false\", accent = \"false\", fo nt_style_name = \"2D Comment\", size = \"12\", foreground = \"[0,0,0] \", background = \"[255,255,255]\"), Typesetting:-mrow(Typesetting:-mi (\"\"), Typesetting:-msup(Typesetting:-mi(\"x\"), Typesetting:-mn(\"2 \"), superscriptshift = \"0\"), Typesetting:-mi(\"\")), Typesetting:-m o(\"−\", form = \"infix\", fence = \"false\", separator = \"fals e\", lspace = \"mediummathspace\", rspace = \"mediummathspace\", stret chy = \"false\", symmetric = \"false\", maxsize = \"infinity\", minsiz e = \"1\", largeop = \"false\", movablelimits = \"false\", accent = \" false\", font_style_name = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Typesetting:-mi(\"x\")) , Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI (_syslibGF'6%-I#miGF$6#Q!F'-F#6(F+-F#6%F+-I%msupGF$6%-F,6#Q\"xF'-I#mnG F$6#Q\"3F'/%1superscriptshiftGQ\"0F'F+-I#moGF$63Q(−F'/%%formGQ&i nfixF'/%&fenceGQ&falseF'/%*separatorGFI/%'lspaceGQ0mediummathspaceF'/% 'rspaceGFN/%)stretchyGFI/%*symmetricGFI/%(maxsizeGQ)infinityF'/%(minsi zeGQ\"1F'/%(largeopGFI/%.movablelimitsGFI/%'accentGFI/%0font_style_nam eGQ+2D~CommentF'/%%sizeGQ#12F'/%+foregroundGQ([0,0,0]F'/%+backgroundGQ .[255,255,255]F'-F#6%F+-F46%F6-F:6#Q\"2F'F=F+F@F6F+" }{TEXT 217 4 ") + " }{XPPEDIT 217 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetti ng:-msub(Typesetting:-mi(\"a\"), Typesetting:-mn(\"4\"), subscriptshif t = \"0\", placeholder = \"false\"), Typesetting:-mi(\"\"));" "-I%mrow G6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-I%msubG F$6&-F,6#Q\"aF'-I#mnGF$6#Q\"4F'/%/subscriptshiftGQ\"0F'/%,placeholderG Q&falseF'F+" }{TEXT 217 6 "(1) + " }{XPPEDIT 217 0 "Typesetting:-mrow( Typesetting:-mi(\"\"), Typesetting:-msub(Typesetting:-mi(\"a\"), Types etting:-mn(\"5\"), subscriptshift = \"0\", placeholder = \"false\"), T ypesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_s yslibGF'6%-I#miGF$6#Q!F'-I%msubGF$6&-F,6#Q\"aF'-I#mnGF$6#Q\"5F'/%/subs criptshiftGQ\"0F'/%,placeholderGQ&falseF'F+" }{TEXT 217 1 "(" } {XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-m row(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mi(\"\"), Ty pesetting:-msup(Typesetting:-mi(\"x\"), Typesetting:-mn(\"3\"), supers criptshift = \"0\"), Typesetting:-mi(\"\")), Typesetting:-mo(\"− \", form = \"infix\", fence = \"false\", separator = \"false\", lspace = \"mediummathspace\", rspace = \"mediummathspace\", stretchy = \"fal se\", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", \+ largeop = \"false\", movablelimits = \"false\", accent = \"false\", fo nt_style_name = \"2D Comment\", size = \"12\", foreground = \"[0,0,0] \", background = \"[255,255,255]\"), Typesetting:-mrow(Typesetting:-mi (\"\"), Typesetting:-msup(Typesetting:-mi(\"x\"), Typesetting:-mn(\"2 \"), superscriptshift = \"0\"), Typesetting:-mi(\"\")), Typesetting:-m o(\"−\", form = \"infix\", fence = \"false\", separator = \"fals e\", lspace = \"mediummathspace\", rspace = \"mediummathspace\", stret chy = \"false\", symmetric = \"false\", maxsize = \"infinity\", minsiz e = \"1\", largeop = \"false\", movablelimits = \"false\", accent = \" false\", font_style_name = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Typesetting:-mi(\"x\"), Typesetting:-mo(\"−\", form = \"infix\", fence = \"false\", sep arator = \"false\", lspace = \"mediummathspace\", rspace = \"mediummat hspace\", stretchy = \"false\", symmetric = \"false\", maxsize = \"inf inity\", minsize = \"1\", largeop = \"false\", movablelimits = \"false \", accent = \"false\", font_style_name = \"2D Comment\", size = \"12 \", foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Typeset ting:-mn(\"1\")), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6 \"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-F#6*F+-F#6%F+-I%msupGF$6% -F,6#Q\"xF'-I#mnGF$6#Q\"3F'/%1superscriptshiftGQ\"0F'F+-I#moGF$63Q(&mi nus;F'/%%formGQ&infixF'/%&fenceGQ&falseF'/%*separatorGFI/%'lspaceGQ0me diummathspaceF'/%'rspaceGFN/%)stretchyGFI/%*symmetricGFI/%(maxsizeGQ)i nfinityF'/%(minsizeGQ\"1F'/%(largeopGFI/%.movablelimitsGFI/%'accentGFI /%0font_style_nameGQ+2D~CommentF'/%%sizeGQ#12F'/%+foregroundGQ([0,0,0] F'/%+backgroundGQ.[255,255,255]F'-F#6%F+-F46%F6-F:6#Q\"2F'F=F+F@F6F@-F :6#FZF+" }{TEXT 217 5 ") = 0" }{TEXT 217 0 "" }}{PARA 208 "" 0 "" {TEXT 204 0 "" }}{PARA 208 "" 0 "" {TEXT 204 17 "We get the system" } {TEXT 204 0 "" }}{PARA 208 "" 0 "" {TEXT 204 0 "" }}{PARA 213 "" 0 "" {TEXT 208 12 " " }{XPPEDIT 208 0 "Typesetting:-mrow(Typeset ting:-mi(\"\"), Typesetting:-msub(Typesetting:-mi(\"a\"), Typesetting: -mn(\"2\"), subscriptshift = \"0\", placeholder = \"false\"), Typesett ing:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF '6%-I#miGF$6#Q!F'-I%msubGF$6&-F,6#Q\"aF'-I#mnGF$6#Q\"2F'/%/subscriptsh iftGQ\"0F'/%,placeholderGQ&falseF'F+" }{TEXT 208 10 " + " } {XPPEDIT 208 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:- msub(Typesetting:-mi(\"a\"), Typesetting:-mn(\"4\"), subscriptshift = \+ \"0\", placeholder = \"false\"), Typesetting:-mi(\"\"));" "-I%mrowG6#/ I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-I%msubGF$6& -F,6#Q\"aF'-I#mnGF$6#Q\"4F'/%/subscriptshiftGQ\"0F'/%,placeholderGQ&fa lseF'F+" }{TEXT 208 3 " - " }{XPPEDIT 208 0 "Typesetting:-mrow(Typeset ting:-mi(\"\"), Typesetting:-msub(Typesetting:-mi(\"a\"), Typesetting: -mn(\"5\"), subscriptshift = \"0\", placeholder = \"false\"), Typesett ing:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF '6%-I#miGF$6#Q!F'-I%msubGF$6&-F,6#Q\"aF'-I#mnGF$6#Q\"5F'/%/subscriptsh iftGQ\"0F'/%,placeholderGQ&falseF'F+" }{TEXT 208 4 " = 0" }{TEXT 208 0 "" }}{PARA 213 "" 0 "" {TEXT 208 3 "- 2" }{XPPEDIT 208 0 "Typesettin g:-mrow(Typesetting:-mi(\"\"), Typesetting:-msub(Typesetting:-mi(\"a\" ), Typesetting:-mn(\"1\"), subscriptshift = \"0\", placeholder = \"fal se\"), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,Typesett ingGI(_syslibGF'6%-I#miGF$6#Q!F'-I%msubGF$6&-F,6#Q\"aF'-I#mnGF$6#Q\"1F '/%/subscriptshiftGQ\"0F'/%,placeholderGQ&falseF'F+" }{TEXT 208 4 " - \+ 3" }{XPPEDIT 208 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetti ng:-msub(Typesetting:-mi(\"a\"), Typesetting:-mn(\"2\"), subscriptshif t = \"0\", placeholder = \"false\"), Typesetting:-mi(\"\"));" "-I%mrow G6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-I%msubG F$6&-F,6#Q\"aF'-I#mnGF$6#Q\"2F'/%/subscriptshiftGQ\"0F'/%,placeholderG Q&falseF'F+" }{TEXT 208 4 " - " }{XPPEDIT 208 0 "Typesetting:-mrow(Ty pesetting:-mi(\"\"), Typesetting:-msub(Typesetting:-mi(\"a\"), Typeset ting:-mn(\"3\"), subscriptshift = \"0\", placeholder = \"false\"), Typ esetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_sys libGF'6%-I#miGF$6#Q!F'-I%msubGF$6&-F,6#Q\"aF'-I#mnGF$6#Q\"3F'/%/subscr iptshiftGQ\"0F'/%,placeholderGQ&falseF'F+" }{TEXT 208 10 " - " }{XPPEDIT 208 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting: -msub(Typesetting:-mi(\"a\"), Typesetting:-mn(\"5\"), subscriptshift = \"0\", placeholder = \"false\"), Typesetting:-mi(\"\"));" "-I%mrowG6# /I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-I%msubGF$6 &-F,6#Q\"aF'-I#mnGF$6#Q\"5F'/%/subscriptshiftGQ\"0F'/%,placeholderGQ&f alseF'F+" }{TEXT 208 4 " = 0" }{TEXT 208 0 "" }}{PARA 213 "" 0 "" {TEXT 208 4 " " }{XPPEDIT 208 0 "Typesetting:-mrow(Typesetting:-mi( \"\"), Typesetting:-msub(Typesetting:-mi(\"a\"), Typesetting:-mn(\"1\" ), subscriptshift = \"0\", placeholder = \"false\"), Typesetting:-mi( \"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#mi GF$6#Q!F'-I%msubGF$6&-F,6#Q\"aF'-I#mnGF$6#Q\"1F'/%/subscriptshiftGQ\"0 F'/%,placeholderGQ&falseF'F+" }{TEXT 208 12 " - " }{XPPEDIT 208 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-msub(Type setting:-mi(\"a\"), Typesetting:-mn(\"3\"), subscriptshift = \"0\", pl aceholder = \"false\"), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulen ameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-I%msubGF$6&-F,6#Q\"a F'-I#mnGF$6#Q\"3F'/%/subscriptshiftGQ\"0F'/%,placeholderGQ&falseF'F+" }{TEXT 208 10 " - " }{XPPEDIT 208 0 "Typesetting:-mrow(Typesett ing:-mi(\"\"), Typesetting:-msub(Typesetting:-mi(\"a\"), Typesetting:- mn(\"5\"), subscriptshift = \"0\", placeholder = \"false\"), Typesetti ng:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF' 6%-I#miGF$6#Q!F'-I%msubGF$6&-F,6#Q\"aF'-I#mnGF$6#Q\"5F'/%/subscriptshi ftGQ\"0F'/%,placeholderGQ&falseF'F+" }{TEXT 208 4 " = 0" }{TEXT 208 0 "" }}{PARA 213 "" 0 "" {TEXT 208 12 " " }{XPPEDIT 208 0 "Ty pesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-msub(Typesetting:- mi(\"a\"), Typesetting:-mn(\"2\"), subscriptshift = \"0\", placeholder = \"false\"), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I, TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-I%msubGF$6&-F,6#Q\"aF'-I#mnGF $6#Q\"2F'/%/subscriptshiftGQ\"0F'/%,placeholderGQ&falseF'F+" }{TEXT 208 3 " + " }{XPPEDIT 208 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), \+ Typesetting:-msub(Typesetting:-mi(\"a\"), Typesetting:-mn(\"3\"), subs criptshift = \"0\", placeholder = \"false\"), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F '-I%msubGF$6&-F,6#Q\"aF'-I#mnGF$6#Q\"3F'/%/subscriptshiftGQ\"0F'/%,pla ceholderGQ&falseF'F+" }{TEXT 208 10 " + " }{XPPEDIT 208 0 "Type setting:-mrow(Typesetting:-mi(\"\"), Typesetting:-msub(Typesetting:-mi (\"a\"), Typesetting:-mn(\"5\"), subscriptshift = \"0\", placeholder = \"false\"), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,Ty pesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-I%msubGF$6&-F,6#Q\"aF'-I#mnGF$6 #Q\"5F'/%/subscriptshiftGQ\"0F'/%,placeholderGQ&falseF'F+" }{TEXT 208 4 " = 0" }{TEXT 208 0 "" }}{PARA 208 "" 0 "" {TEXT 204 0 "" }}{PARA 208 "" 0 "" {TEXT 204 41 "The coefficient matrix for the system is:" } {TEXT 204 0 "" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 221 73 "AP := \+ Matrix([[0,1,0,1,-1], [-2,-3,-1,0,-1], [1,0,-1,0,-1],[0,1,1,0,1]]);" } {MPLTEXT 1 221 0 "" }}}{EXCHG {PARA 208 "" 0 "" {TEXT 204 133 "Note th at the columns of this matrix correspond to the coordinates of the ind ividual vectors (with respect to the standard basis for " }{XPPEDIT 204 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-msub(Type setting:-mi(\"P\"), Typesetting:-mn(\"3\"), subscriptshift = \"0\", pl aceholder = \"false\"), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulen ameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-I%msubGF$6&-F,6#Q\"P F'-I#mnGF$6#Q\"3F'/%/subscriptshiftGQ\"0F'/%,placeholderGQ&falseF'F+" }{TEXT 204 2 ")." }{TEXT 204 0 "" }}{PARA 208 "" 0 "" {TEXT 204 35 "Re duce to reduced row echelon form:" }{TEXT 204 0 "" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 221 32 "BP := ReducedRowEchelonForm(AP);" } {MPLTEXT 1 221 0 "" }}}{EXCHG {PARA 208 "" 0 "" {TEXT 204 72 "Note tha t according to (a) the rank of BP, and hence of AP, is 3. Since " } {XPPEDIT 204 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:- msub(Typesetting:-mi(\"P\"), Typesetting:-mn(\"3\"), subscriptshift = \+ \"0\", placeholder = \"false\"), Typesetting:-mi(\"\"));" "-I%mrowG6#/ I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-I%msubGF$6& -F,6#Q\"PF'-I#mnGF$6#Q\"3F'/%/subscriptshiftGQ\"0F'/%,placeholderGQ&fa lseF'F+" }{TEXT 204 564 " has dimension 4, and not every polynomial sa tisfies the restriction specified (that the value of the derivative be zero at x=1), our subspace is proper. We conclude that the subspace a ctually has dimension 3, and that the polynomials we listed do span th e subspace. Now according to (c), we can choose the first three polyno mials listed as our basis (they are linearly independent), and the fou rth and fifth polymomials can be expressed as linear combinations of t he first three with coefficients given by the entries of BP in those l ast two columns. Let's check:" }{TEXT 204 0 "" }}}{EXCHG {PARA 216 "> \+ " 0 "" {MPLTEXT 1 221 14 "p1 := x^2-2*x;" }{MPLTEXT 1 221 0 "" } {MPLTEXT 1 221 2 "\n" }{MPLTEXT 1 221 16 "p2 := x^3-3*x+1;" }{MPLTEXT 1 221 0 "" }{MPLTEXT 1 221 2 "\n" }{MPLTEXT 1 221 16 "p3 := x^3-x^2-x; " }{MPLTEXT 1 221 0 "" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 221 37 "BP[1,4]*p1 + BP[2,4]*p2 + BP[3,4]*p3;" }{MPLTEXT 1 221 0 "" } {MPLTEXT 1 221 2 "\n" }{MPLTEXT 1 221 37 "BP[1,5]*p1 + BP[2,5]*p2 + BP [3,5]*p3;" }{MPLTEXT 1 221 0 "" }}}{EXCHG {PARA 208 "" 0 "" {TEXT 204 66 "These are exactly the remaining two polynomials we listed earlier. " }{TEXT 204 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 227 8 "Exercise" }} {EXCHG {PARA 208 "" 0 "" {TEXT 204 29 "2. Let W be the subspace of " }{XPPEDIT 204 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting: -msub(Typesetting:-mi(\"M\"), Typesetting:-mrow(Typesetting:-mn(\"2\") , Typesetting:-mo(\"⁢\", form = \"infix\", fence = \"fa lse\", separator = \"false\", lspace = \"0em\", rspace = \"0em\", stre tchy = \"false\", symmetric = \"false\", maxsize = \"infinity\", minsi ze = \"1\", largeop = \"false\", movablelimits = \"false\", accent = \+ \"false\", font_style_name = \"_cstyle10\", size = \"12\", foreground \+ = \"[0,0,0]\", background = \"[255,255,255]\"), Typesetting:-mi(\"x2\" )), subscriptshift = \"0\", placeholder = \"false\"), Typesetting:-mi( \"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#mi GF$6#Q!F'-I%msubGF$6&-F,6#Q\"MF'-F#6%-I#mnGF$6#Q\"2F'-I#moGF$63Q1&Invi sibleTimes;F'/%%formGQ&infixF'/%&fenceGQ&falseF'/%*separatorGFD/%'lspa ceGQ$0emF'/%'rspaceGFI/%)stretchyGFD/%*symmetricGFD/%(maxsizeGQ)infini tyF'/%(minsizeGQ\"1F'/%(largeopGFD/%.movablelimitsGFD/%'accentGFD/%0fo nt_style_nameGQ*_cstyle10F'/%%sizeGQ#12F'/%+foregroundGQ([0,0,0]F'/%+b ackgroundGQ.[255,255,255]F'-F,6#Q#x2F'/%/subscriptshiftGQ\"0F'/%,place holderGFDF+" }{TEXT 204 211 "(R) consisting of the symmetric 2x2 matri ces. The set S listed below generates W. Find a subset of S that is a \+ basis for W, and write the other matrices in S as linear combinations \+ of the elements of your basis." }}{PARA 0 "" 0 "" {TEXT 213 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 221 32 "M1 := Matrix([[0,-1],[-1,1 ]]);\n" }{MPLTEXT 1 221 30 "M2 := Matrix([[1,2],[2,3]]);\n" }{MPLTEXT 1 221 30 "M3 := Matrix([[2,1],[1,9]]);\n" }{MPLTEXT 1 221 32 "M4 := Ma trix([[1,-2],[-2,4]]);\n" }{MPLTEXT 1 221 30 "M5 := Matrix([[-1,2],[2, -1]]);" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "" "%#%?G" }}}} {EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 221 0 "" }}}}{PARA 205 "" 0 "" {TEXT 209 0 "" }}{PARA 205 "" 0 "" {TEXT 228 0 "" }}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 15 10 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }