(*^
::[	Information =

	"This is a Mathematica Notebook file.  It contains ASCII text, and can be
	transferred by email, ftp, or other text-file transfer utility.  It should
	be read or edited using a copy of Mathematica or MathReader.  If you 
	received this as email, use your mail application or copy/paste to save 
	everything from the line containing (*^ down to the line containing ^*)
	into a plain text file.  On some systems you may have to give the file a 
	name ending with ".ma" to allow Mathematica to recognize it as a Notebook.
	The line below identifies what version of Mathematica created this file,
	but it can be opened using any other version as well.";

	FrontEndVersion = "Macintosh Mathematica Notebook Front End Version 2.2";

	MacintoshStandardFontEncoding; 
	
	fontset = title, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, e8,  24, "Times"; 
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	fontset = subsubtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, italic, e6,  14, "Times"; 
	fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M22, bold, a20,  18, "Times"; 
	fontset = subsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, blackBox, M19, bold, a15,  14, "Times"; 
	fontset = subsubsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, whiteBox, M18, bold, a12,  12, "Times"; 
	fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7,  12, "Times"; 
	fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7,  10, "Times"; 
	fontset = input, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeInput, M42, N23, bold, L-5,  12, "Courier"; 
	fontset = output, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5,  12, "Courier"; 
	fontset = message, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, R65535, L-5,  12, "Courier"; 
	fontset = print, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5,  12, "Courier"; 
	fontset = info, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, B65535, L-5,  12, "Courier"; 
	fontset = postscript, PostScript, formatAsPostScript, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeGraphics, M7, l34, w282, h287,  12, "Courier"; 
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	fontset = leftheader, inactive, L2,  12, "Times"; 
	fontset = footer, inactive, noKeepOnOnePage, preserveAspect, center, M7,  12, "Times"; 
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	fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7,  10, "Times"; 
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	fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7,  12, "Times"; 
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	fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7,  12, "Times"; 
	paletteColors = 128; automaticGrouping; currentKernel; 
]
:[font = input; initialization; preserveAspect]
*)
$Post:=If[MatrixQ[#],MatrixForm[#],#]&
(*
:[font = input; initialization; preserveAspect]
*)
Exercise 2.4.1
a)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
Do[v[i]=Table[i+j-1,{j,5}],{i,4}]
RowReduce[Table[v[i],{i,4}]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1, 0, -1, -2, -3}, {0, 1, 2, 3, 4}, 
 
   {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}}]
;[o]
1    0    -1   -2   -3

0    1    2    3    4

0    0    0    0    0

0    0    0    0    0
:[font = input; initialization; preserveAspect; startGroup]
*)
Array[%[[#]]&,2]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1, 0, -1, -2, -3}, {0, 1, 2, 3, 4}}]
;[o]
1    0    -1   -2   -3

0    1    2    3    4
:[font = input; initialization; preserveAspect]
*)
b)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
Do[v[i]=Table[1/(i+j),{j,6}],{i,7}]
RowReduce[Table[v[i],{i,7}]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1, 0, 0, 0, 0, 0}, {0, 1, 0, 0, 0, 0}, 
 
   {0, 0, 1, 0, 0, 0}, {0, 0, 0, 1, 0, 0}, 
 
   {0, 0, 0, 0, 1, 0}, {0, 0, 0, 0, 0, 1}, 
 
   {0, 0, 0, 0, 0, 0}}]
;[o]
1   0   0   0   0   0

0   1   0   0   0   0

0   0   1   0   0   0

0   0   0   1   0   0

0   0   0   0   1   0

0   0   0   0   0   1

0   0   0   0   0   0
:[font = input; initialization; preserveAspect; startGroup]
*)
Array[%[[#]]&,6]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1, 0, 0, 0, 0, 0}, {0, 1, 0, 0, 0, 0}, 
 
   {0, 0, 1, 0, 0, 0}, {0, 0, 0, 1, 0, 0}, 
 
   {0, 0, 0, 0, 1, 0}, {0, 0, 0, 0, 0, 1}}]
;[o]
1   0   0   0   0   0

0   1   0   0   0   0

0   0   1   0   0   0

0   0   0   1   0   0

0   0   0   0   1   0

0   0   0   0   0   1
:[font = input; initialization; preserveAspect]
*)
c)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
A=Table[1/(i+j-1),{i,7},{j,9}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9}, 
 
   {1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9, 1/10}, 
 
   {1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9, 1/10, 1/11}, 
 
   {1/4, 1/5, 1/6, 1/7, 1/8, 1/9, 1/10, 1/11, 1/12}, 
 
   {1/5, 1/6, 1/7, 1/8, 1/9, 1/10, 1/11, 1/12, 1/13}, 
 
   {1/6, 1/7, 1/8, 1/9, 1/10, 1/11, 1/12, 1/13, 1/14}, 
 
   {1/7, 1/8, 1/9, 1/10, 1/11, 1/12, 1/13, 1/14, 1/15}}]
;[o]
     1    1    1    1    1    1    1    1
     -    -    -    -    -    -    -    -
1    2    3    4    5    6    7    8    9

1    1    1    1    1    1    1    1    1
-    -    -    -    -    -    -    -    --
2    3    4    5    6    7    8    9    10

1    1    1    1    1    1    1    1    1
-    -    -    -    -    -    -    --   --
3    4    5    6    7    8    9    10   11

1    1    1    1    1    1    1    1    1
-    -    -    -    -    -    --   --   --
4    5    6    7    8    9    10   11   12

1    1    1    1    1    1    1    1    1
-    -    -    -    -    --   --   --   --
5    6    7    8    9    10   11   12   13

1    1    1    1    1    1    1    1    1
-    -    -    -    --   --   --   --   --
6    7    8    9    10   11   12   13   14

1    1    1    1    1    1    1    1    1
-    -    -    --   --   --   --   --   --
7    8    9    10   11   12   13   14   15
:[font = input; initialization; preserveAspect; startGroup]
*)
NullSpace[A]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{-7/6435, 128/2145, -112/143, 1792/429, -140/13, 
 
    896/65, -112/15, 0, 1}, 
 
   {-1/3432, 7/429, -63/286, 175/143, -175/52, 63/13, -7/2, 
 
    1, 0}}]
;[o]
   7      128         112     1792        140     896
-(----)   ----      -(---)    ----      -(---)    ---
  6435    2145        143     429         13      65
 
        112
      -(---)
        15      0         1

   1       7          63      175         175     63
-(----)   ---       -(---)    ---       -(---)    --
  3432    429         286     143         52      13
 
        7
      -(-)
        2       1         0
:[font = input; initialization; preserveAspect]
*)
Exercise 2.4.2
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
Do[q[i]=Expand[Sum[(x-i)^j,{j,0,12}]],{i,13}]
A=Table[Coefficient[q[i],x,j-1],{j,13},{i,13}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1, 2731, 398581, 13421773, 203450521, 
 
    1865813431, 12111126301, 61083979321, 254186582833, 
 
    909090909091, 2876892678661, 8230246567621, 
 
    21633936185161}, {-6, -16839, -1627538, -40936407, 
 
    -495062934, -3776050991, -20978200914, -92474357583, 
 
    -341739739142, -1099173553719, -3160223018226, 
 
    -8283004558439, -20088655029078}, 
 
   {36, 47635, 3047482, 57243861, 552255136, 3503159911, 
 
    16656660630, 64171171177, 210598291036, 609166040571, 
 
    1591173940786, 3820896027325, 8549989693272}, 
 
   {-125, -81743, -3460133, -48529775, -373452269, 
 
    -1970023775, -8016407573, -26990998943, -78662204765, 
 
    -204621269039, -485577457925, -1068266933903, 
 
    -2205538498253}, {295, 94779, 2653303, 27780637, 
 
    170505351, 747932695, 2604552679, 7663852953, 
 
    19834344487, 46398066451, 100029595095, 201613539829, 
 
    384048468103}, {-496, -78231, -1447676, -11312839, 
 
    -55371504, -201961871, -601843276, -1547602671, 
 
    -3556678544, -7481993959, -14654190396, -27059454071, 
 
    -47556926416}, {610, 47139, 576304, 3360421, 13115166, 
 
    39772615, 101419564, 227901129, 465089626, 879818731, 
 
    1565482920, 2648302189, 4294252054}, 
 
   {-553, -20895, -168665, -733663, -2282889, -5755583, 
 
    -12558265, -24659775, -44686313, -76016479, -122876313, 
 
    -190434335, -284896585}, 
 
   {367, 6763, 36019, 116845, 289831, 607447, 1134043, 
 
    1945849, 3130975, 4789411, 7033027, 9985573, 13782679}, 
 
   {-174, -1559, -5474, -13239, -26174, -45599, -72834, 
 
    -109199, -156014, -214599, -286274, -372359, -474174}, 
 
   {56, 243, 562, 1013, 1596, 2311, 3158, 4137, 5248, 6491, 
 
    7866, 9373, 11012}, {-11, -23, -35, -47, -59, -71, -83, 
 
    -95, -107, -119, -131, -143, -155}, 
 
   {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}}]
;[o]
1                 2731              398581
 
   13421773          203450521         1865813431
 
      12111126301       61083979321       254186582833
 
      909090909091      2876892678661     8230246567621
 
      21633936185161

-6                -16839            -1627538
 
   -40936407         -495062934        -3776050991
 
      -20978200914      -92474357583      -341739739142
 
      -1099173553719    -3160223018226    -8283004558439
 
      -20088655029078

36                47635             3047482
 
   57243861          552255136         3503159911
 
      16656660630       64171171177       210598291036
 
      609166040571      1591173940786     3820896027325
 
      8549989693272

-125              -81743            -3460133
 
   -48529775         -373452269        -1970023775
 
      -8016407573       -26990998943      -78662204765
 
      -204621269039     -485577457925     -1068266933903
 
      -2205538498253

295               94779             2653303
 
   27780637          170505351         747932695
 
      2604552679        7663852953        19834344487
 
      46398066451       100029595095      201613539829
 
      384048468103

-496              -78231            -1447676
 
   -11312839         -55371504         -201961871
 
      -601843276        -1547602671       -3556678544
 
      -7481993959       -14654190396      -27059454071
 
      -47556926416

610               47139             576304
 
   3360421           13115166          39772615
 
      101419564         227901129         465089626
 
      879818731         1565482920        2648302189
 
      4294252054

-553              -20895            -168665
 
   -733663           -2282889          -5755583
 
      -12558265         -24659775         -44686313
 
      -76016479         -122876313        -190434335
 
      -284896585

367               6763              36019
 
   116845            289831            607447
 
      1134043           1945849           3130975
 
      4789411           7033027           9985573
 
      13782679

-174              -1559             -5474
 
   -13239            -26174            -45599
 
      -72834            -109199           -156014
 
      -214599           -286274           -372359
 
      -474174

56                243               562
 
   1013              1596              2311
 
      3158              4137              5248
 
      6491              7866              9373
 
      11012

-11               -23               -35
 
   -47               -59               -71
 
      -83               -95               -107
 
      -119              -131              -143
 
      -155

1                 1                 1
 
   1                 1                 1
 
      1                 1                 1
 
      1                 1                 1
 
      1
:[font = input; initialization; preserveAspect; startGroup]
*)
b=Table[Coefficient[Sum[x^j,{j,0,12}],x,i-1],{i,13}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}
;[o]
{1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}
:[font = input; initialization; preserveAspect]
*)
(which was obvious, could have been entered by hand)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
LinearSolve[A,b]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{13, -78, 286, -715, 1287, -1716, 1716, -1287, 715, -286, 
 
  78, -13, 1}
;[o]
{13, -78, 286, -715, 1287, -1716, 1716, -1287, 715, -286, 
 
  78, -13, 1}
:[font = input; initialization; preserveAspect]
*)
The above solution indicates the coeff's of a linear 
combination of q[i]'s which gives p.
(*
:[font = input; initialization; preserveAspect]
*)
Exercise 2.4.3
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
NullSpace[A]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{}
;[o]
{}
:[font = input; initialization; preserveAspect]
*)
The 13 linearly independent vectors form a basis in R^13,
and therefore the q[i]'s fprm a basis in P12.
(*
:[font = input; initialization; preserveAspect]
*)
Exercise 2.4.4
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
B=Array[Max,{5,5}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1, 2, 3, 4, 5}, {2, 2, 3, 4, 5}, 
 
   {3, 3, 3, 4, 5}, {4, 4, 4, 4, 5}, {5, 5, 5, 5, 5}}]
;[o]
1   2   3   4   5

2   2   3   4   5

3   3   3   4   5

4   4   4   4   5

5   5   5   5   5
:[font = input; initialization; preserveAspect; startGroup]
*)
Unprotect[C]; C=Array[Min,{5,5}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1, 1, 1, 1, 1}, {1, 2, 2, 2, 2}, 
 
   {1, 2, 3, 3, 3}, {1, 2, 3, 4, 4}, {1, 2, 3, 4, 5}}]
;[o]
1   1   1   1   1

1   2   2   2   2

1   2   3   3   3

1   2   3   4   4

1   2   3   4   5
:[font = input; initialization; preserveAspect]
*)
Clear[b]
Do[b[i]=B[[i]],{i,5}]
Do[c[i]=C[[i]],{i,5}]
(*
:[font = input; initialization; preserveAspect]
*)
The above is correct due to the symmetry of B and C.
For that reason, B={b[1], ..., b[5]} and C={c[1], ..., c[5]}.
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
v={5,4,3,2,1}
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{5, 4, 3, 2, 1}
;[o]
{5, 4, 3, 2, 1}
:[font = input; initialization; preserveAspect]
*)
a)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
LinearSolve[B,v]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{-1, 0, 0, 0, 6/5}
;[o]
              6
{-1, 0, 0, 0, -}
              5
:[font = input; initialization; preserveAspect]
*)
b)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
LinearSolve[C,v]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{6, 0, 0, 0, -1}
;[o]
{6, 0, 0, 0, -1}
:[font = input; initialization; preserveAspect]
*)
(Given that b[1], ..., b[5] and c[1], ..., c[5] are bases
for R^5, uniqueness of solutions is guarranteed.)
(*
:[font = input; initialization; preserveAspect]
*)
c)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
P=Inverse[C].B
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{0, 2, 3, 4, 5}, {0, -1, 0, 0, 0}, 
 
   {0, 0, -1, 0, 0}, {0, 0, 0, -1, 0}, {1, 1, 1, 1, 0}}]
;[o]
0    2    3    4    5

0    -1   0    0    0

0    0    -1   0    0

0    0    0    -1   0

1    1    1    1    0
:[font = input; initialization; preserveAspect]
*)
d)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
Q=Inverse[B].C
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{0, 1, 1, 1, 1}, {0, -1, 0, 0, 0}, 
 
   {0, 0, -1, 0, 0}, {0, 0, 0, -1, 0}, 
 
   {1/5, 2/5, 3/5, 4/5, 0}}]
;[o]


0    1    1    1    1



0    -1   0    0    0



0    0    -1   0    0



0    0    0    -1   0

1    2    3    4
-    -    -    -
5    5    5    5    0
:[font = input; initialization; preserveAspect]
*)
e)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
P.LinearSolve[B,v]-LinearSolve[C,v]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{0, 0, 0, 0, 0}
;[o]
{0, 0, 0, 0, 0}
:[font = input; initialization; preserveAspect]
*)
f)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
Q.LinearSolve[C,v]-LinearSolve[B,v]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{0, 0, 0, 0, 0}
;[o]
{0, 0, 0, 0, 0}
:[font = input; initialization; preserveAspect]
*)
g)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
P.Q
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1, 0, 0, 0, 0}, {0, 1, 0, 0, 0}, 
 
   {0, 0, 1, 0, 0}, {0, 0, 0, 1, 0}, {0, 0, 0, 0, 1}}]
;[o]
1   0   0   0   0

0   1   0   0   0

0   0   1   0   0

0   0   0   1   0

0   0   0   0   1
^*)