(*^
::[	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"; 
	fontset = subtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, e6,  18, "Times"; 
	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"; 
	fontset = name, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, italic,  10, "Geneva"; 
	fontset = header, inactive, noKeepOnOnePage, preserveAspect, M7,  12, "Times"; 
	fontset = leftheader, inactive, L2,  12, "Times"; 
	fontset = footer, inactive, noKeepOnOnePage, preserveAspect, center, M7,  12, "Times"; 
	fontset = leftfooter, inactive, L2,  12, "Times"; 
	fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7,  10, "Times"; 
	fontset = clipboard, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7,  12, "Times"; 
	fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7,  12, "Times"; 
	fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7,  12, "Times"; 
	fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7,  12, "Times"; 
	fontset = special3, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7,  12, "Times"; 
	fontset = special4, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7,  12, "Times"; 
	fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7,  12, "Times"; 
	paletteColors = 128; automaticGrouping; currentKernel; 
]
:[font = input; initialization; preserveAspect]
*)
Exercise 4.1.1
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
T[x_]:=Array[Sum[x[[j]],{j,6-#}]&,4]
U=Table[Subscripted[u[i]],{i,5}];
U1=Table[Subscripted[u1[i]],{i,5}];
U2=Table[Subscripted[u2[i]],{i,5}];
T[U]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{Subscripted[u[1]] + Subscripted[u[2]] + Subscripted[u[3]] + 
 
   Subscripted[u[4]] + Subscripted[u[5]], 
 
  Subscripted[u[1]] + Subscripted[u[2]] + 
 
   Subscripted[u[3]] + Subscripted[u[4]], 
 
  Subscripted[u[1]] + Subscripted[u[2]] + Subscripted[u[3]], 
 
  Subscripted[u[1]] + Subscripted[u[2]]}
;[o]
{u  + u  + u  + u  + u , u  + u  + u  + u , u  + u  + u , 
  1    2    3    4    5   1    2    3    4   1    2    3
 
  u  + u }
   1    2
:[font = input; initialization; preserveAspect]
*)
a)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
T[U1+U2]-T[U1]-T[U2]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{0, 0, 0, 0}
;[o]
{0, 0, 0, 0}
:[font = input; initialization; preserveAspect; startGroup]
*)
T[k U]-k T[U]//Simplify
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{0, 0, 0, 0}
;[o]
{0, 0, 0, 0}
:[font = input; initialization; preserveAspect]
*)
b)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
e=IdentityMatrix[5]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{{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}}
:[font = input; initialization; preserveAspect; startGroup]
*)
Do[Print[T[e[[i]]]],{i,5}]
(*
:[font = print; inactive; preserveAspect; endGroup]
{1, 1, 1, 1}
{1, 1, 1, 1}
{1, 1, 1, 0}
{1, 1, 0, 0}
{1, 0, 0, 0}
:[font = input; initialization; preserveAspect]
*)
Exercise 4.1.2
(*
:[font = input; initialization; preserveAspect]
*)
U={{1,1,1,0},
   {1,1,0,0},
   {1,0,0,0}};
(*
:[font = input; initialization; preserveAspect]
*)
a)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
NullSpace[U]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{{0, 0, 0, 1}}
;[o]
{{0, 0, 0, 1}}
:[font = input; initialization; preserveAspect; startGroup]
*)
Clear[T]
<<LinearAlgebra`Master`
T[v_]:=v-Projection[v,NullSpace[U][[1]]]
V=Table[Subscripted[v[i]],{i,4}];
T[V]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{Subscripted[v[1]], Subscripted[v[2]], Subscripted[v[3]], 0}
;[o]
{v , v , v , 0}
  1   2   3
:[font = input; initialization; preserveAspect]
*)
b)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
V1=Table[Subscripted[v1[i]],{i,4}];
V2=Table[Subscripted[v2[i]],{i,4}];
T[V1+V2]-T[V1]-T[V2]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{0, 0, 0, 0}
;[o]
{0, 0, 0, 0}
:[font = input; initialization; preserveAspect; startGroup]
*)
T[k V]-k T[V]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{0, 0, 0, 0}
;[o]
{0, 0, 0, 0}
:[font = input; initialization; preserveAspect]
*)
c)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
T[T[V]]-T[V]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{0, 0, 0, 0}
;[o]
{0, 0, 0, 0}
:[font = input; initialization; preserveAspect]
*)
d)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
u=Sum[K[i]U[[i]],{i,3}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{K[1] + K[2] + K[3], K[1] + K[2], K[1], 0}
;[o]
{K[1] + K[2] + K[3], K[1] + K[2], K[1], 0}
:[font = input; initialization; preserveAspect; startGroup]
*)
T[u]-u
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{0, 0, 0, 0}
;[o]
{0, 0, 0, 0}
:[font = input; initialization; preserveAspect]
*)
Exercise 4.1.3
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
M=Array[1/(#1+#2)&,{7,5}]; M//MatrixForm
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1/2, 1/3, 1/4, 1/5, 1/6}, 
 
   {1/3, 1/4, 1/5, 1/6, 1/7}, {1/4, 1/5, 1/6, 1/7, 1/8}, 
 
   {1/5, 1/6, 1/7, 1/8, 1/9}, {1/6, 1/7, 1/8, 1/9, 1/10}, 
 
   {1/7, 1/8, 1/9, 1/10, 1/11}, {1/8, 1/9, 1/10, 1/11, 1/12}}
 
   ]
;[o]
1    1    1    1    1
-    -    -    -    -
2    3    4    5    6

1    1    1    1    1
-    -    -    -    -
3    4    5    6    7

1    1    1    1    1
-    -    -    -    -
4    5    6    7    8

1    1    1    1    1
-    -    -    -    -
5    6    7    8    9

1    1    1    1    1
-    -    -    -    --
6    7    8    9    10

1    1    1    1    1
-    -    -    --   --
7    8    9    10   11

1    1    1    1    1
-    -    --   --   --
8    9    10   11   12
:[font = input; initialization; preserveAspect; startGroup]
*)
Length[NullSpace[M]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
0
;[o]
0
:[font = input; initialization; preserveAspect; startGroup]
*)
B=GramSchmidt[Transpose[M],Normalized->False]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{{1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8}, 
 
  {-39451/1116447, 18847/4465788, 29149/1860745, 
 
   42869/2232894, 156047/7815129, 58549/2977192, 
 
   63449/3349341}, {30896049/28593572036, 
 
   -227275973/107225895135, -51511759/42890358054, 
 
   -5933919/50038751063, 6193793/9029549064, 
 
   79401149/64335537081, 114149009/71483930090}, 
 
  {-332946827/17467678384365, 183932239/1905564914658, 
 
   -1412639483/24454749738111, -2288927063/27948285414984, 
 
   -408756419/10480607030619, 288677241/11645118922910, 
 
   103741447/1164511892291}, 
 
  {2905769/14542895080470, -1077608663/559901460598095, 
 
   184380135/42659158902712, -33917119/47991553765551, 
 
   -181287461/53323948628390, -44407063/26661974314195, 
 
   94704917/29085790160940}}
;[o]
  1  1  1  1  1  1  1       39451     18847    29149
{{-, -, -, -, -, -, -}, {-(-------), -------, -------, 
  2  3  4  5  6  7  8      1116447   4465788  1860745
 
    42869   156047    58549    63449
   -------, -------, -------, -------}, 
   2232894  7815129  2977192  3349341
 
    30896049       227275973        51511759
  {-----------, -(------------), -(-----------), 
   28593572036    107225895135     42890358054
 
       5933919      6193793     79401149     114149009
   -(-----------), ----------, -----------, -----------}, 
     50038751063   9029549064  64335537081  71483930090
 
       332946827        183932239        1412639483
  {-(--------------), -------------, -(--------------), 
     17467678384365   1905564914658    24454749738111
 
       2288927063         408756419        288677241
   -(--------------), -(--------------), --------------, 
     27948285414984     10480607030619   11645118922910
 
     103741447         2905769          1077608663
   -------------}, {--------------, -(---------------), 
   1164511892291    14542895080470    559901460598095
 
     184380135          33917119          181287461
   --------------, -(--------------), -(--------------), 
   42659158902712    47991553765551     53323948628390
 
        44407063         94704917
   -(--------------), --------------}}
     26661974314195   29085790160940
:[font = input; initialization; preserveAspect; startGroup]
*)
Clear[T]
T[v_]:=Sum[Projection[v,B[[i]]],{i,5}]
v={1,2,3,4,5,6,7};
T[v]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{89160409086/89135248205, 177873530092/89135248205, 
 
  53835221103/17827049641, 70771630756/17827049641, 
 
  89151626819/17827049641, 537642502338/89135248205, 
 
  622311857111/89135248205}
;[o]
 89160409086  177873530092  53835221103  70771630756
{-----------, ------------, -----------, -----------, 
 89135248205  89135248205   17827049641  17827049641
 
  89151626819  537642502338  622311857111
  -----------, ------------, ------------}
  17827049641  89135248205   89135248205
:[font = input; initialization; preserveAspect; startGroup]
*)
Clear[v]
V=Table[Subscripted[v[i]],{i,7}];
V1=Table[Subscripted[v1[i]],{i,7}];
V2=Table[Subscripted[v2[i]],{i,7}];
T[V1+V2]-T[V1]-T[V2]//Expand
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{0, 0, 0, 0, 0, 0, 0}
;[o]
{0, 0, 0, 0, 0, 0, 0}
:[font = input; initialization; preserveAspect; startGroup]
*)
T[k V]-k T[V]//Expand
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{0, 0, 0, 0, 0, 0, 0}
;[o]
{0, 0, 0, 0, 0, 0, 0}
:[font = input; initialization; preserveAspect]
*)
Exercise 4.1.4
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
Do[p[i]=(x-2)^(i-1),{i,6}]
B=Transpose[Table[Coefficient[p[i],x,j-1],{i,6},{j,6}]];
B//MatrixForm
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1, -2, 4, -8, 16, -32}, 
 
   {0, 1, -4, 12, -32, 80}, {0, 0, 1, -6, 24, -80}, 
 
   {0, 0, 0, 1, -8, 40}, {0, 0, 0, 0, 1, -10}, 
 
   {0, 0, 0, 0, 0, 1}}]
;[o]
1     -2    4     -8    16    -32

0     1     -4    12    -32   80

0     0     1     -6    24    -80

0     0     0     1     -8    40

0     0     0     0     1     -10

0     0     0     0     0     1
:[font = input; initialization; preserveAspect]
*)
a)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
NullSpace[B]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{}
;[o]
{}
:[font = input; initialization; preserveAspect]
*)
b)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
IB=Inverse[B]; IB//MatrixForm
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1, 2, 4, 8, 16, 32}, {0, 1, 4, 12, 32, 80}, 
 
   {0, 0, 1, 6, 24, 80}, {0, 0, 0, 1, 8, 40}, 
 
   {0, 0, 0, 0, 1, 10}, {0, 0, 0, 0, 0, 1}}]
;[o]
1    2    4    8    16   32

0    1    4    12   32   80

0    0    1    6    24   80

0    0    0    1    8    40

0    0    0    0    1    10

0    0    0    0    0    1
:[font = input; initialization; preserveAspect; startGroup]
*)
Clear[T]
T[p_]:=IB.Table[Coefficient[p,x,i-1],{i,6}]
p0=Sum[Subscripted[a0[i-1]]x^(i-1),{i,6}];
p1=Sum[Subscripted[a1[i-1]]x^(i-1),{i,6}];
p2=Sum[Subscripted[a2[i-1]]x^(i-1),{i,6}];
T[p1+p2]-T[p1]-T[p2]//Simplify
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{0, 0, 0, 0, 0, 0}
;[o]
{0, 0, 0, 0, 0, 0}
:[font = input; initialization; preserveAspect; startGroup]
*)
T[k p0]-k T[p0]//Simplify
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{0, 0, 0, 0, 0, 0}
;[o]
{0, 0, 0, 0, 0, 0}
:[font = input; initialization; preserveAspect]
*)
c)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
Do[Print[T[x^i]],{i,5}]
(*
:[font = print; inactive; preserveAspect; endGroup]
{2, 1, 0, 0, 0, 0}
{4, 4, 1, 0, 0, 0}
{8, 12, 6, 1, 0, 0}
{16, 32, 24, 8, 1, 0}
{32, 80, 80, 40, 10, 1}
:[font = input; initialization; preserveAspect]
*)
d)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
T[Sum[Subscripted[a[i-1]]x^(i-1),{i,6}]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{Subscripted[a[0]] + 2*Subscripted[a[1]] + 
 
   4*Subscripted[a[2]] + 8*Subscripted[a[3]] + 
 
   16*Subscripted[a[4]] + 32*Subscripted[a[5]], 
 
  Subscripted[a[1]] + 4*Subscripted[a[2]] + 
 
   12*Subscripted[a[3]] + 32*Subscripted[a[4]] + 
 
   80*Subscripted[a[5]], 
 
  Subscripted[a[2]] + 6*Subscripted[a[3]] + 
 
   24*Subscripted[a[4]] + 80*Subscripted[a[5]], 
 
  Subscripted[a[3]] + 8*Subscripted[a[4]] + 
 
   40*Subscripted[a[5]], 
 
  Subscripted[a[4]] + 10*Subscripted[a[5]], 
 
  Subscripted[a[5]]}
;[o]
{a  + 2 a  + 4 a  + 8 a  + 16 a  + 32 a , 
  0      1      2      3       4       5
 
  a  + 4 a  + 12 a  + 32 a  + 80 a , 
   1      2       3       4       5
 
  a  + 6 a  + 24 a  + 80 a , a  + 8 a  + 40 a , a  + 10 a , 
   2      3       4       5   3      4       5   4       5
 
  a }
   5
:[font = input; initialization; preserveAspect]
*)
Exercise 4.1.5
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
Clear[u,U]
u[i_]:=Table[Subscripted[a[i,j]],{j,5}]
U[i_]:=Table[Subscripted[b[i,j]],{j,5}]
L[v_]:=Det[{u[1],u[2],u[3],u[4],v}]
L[U[1]+U[2]]-L[U[1]]-L[U[2]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
0
;[o]
0
:[font = input; initialization; preserveAspect; startGroup]
*)
k L[U[0]]-L[k U[0]]//Expand
(*
:[font = output; output; inactive; preserveAspect; endGroup]
0
;[o]
0
^*)