(*^
::[	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.";

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	paletteColors = 128; automaticGrouping; currentKernel; 
]
:[font = input; initialization; preserveAspect]
*)
$Post:=If[MatrixQ[#],MatrixForm[#],#]&
(*
:[font = input; preserveAspect]
Exercise 4.2.1
:[font = input; preserveAspect]
By tabulating the solutions of LinearSolve, we obtain
on the ROWS the solutions X[[i]] of F.X[[i]]=L[b[[i]]];
in the above equation nevertheless, the X[[i]]'s are
column vectors, so if we want to build a matrix with
them in order to preserve the result of F.X, we have
to put the X[[i]] solutions as columnns, hence transpose.
:[font = input; initialization; preserveAspect]
*)
Exercise 4.2.2
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
b=Table[If[i==j,0,1],{i,7},{j,7}];
B=Transpose[b]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{0, 1, 1, 1, 1, 1, 1}, {1, 0, 1, 1, 1, 1, 1}, 
 
   {1, 1, 0, 1, 1, 1, 1}, {1, 1, 1, 0, 1, 1, 1}, 
 
   {1, 1, 1, 1, 0, 1, 1}, {1, 1, 1, 1, 1, 0, 1}, 
 
   {1, 1, 1, 1, 1, 1, 0}}]
;[o]
0   1   1   1   1   1   1

1   0   1   1   1   1   1

1   1   0   1   1   1   1

1   1   1   0   1   1   1

1   1   1   1   0   1   1

1   1   1   1   1   0   1

1   1   1   1   1   1   0
:[font = input; initialization; preserveAspect]
*)
Of course transposing was not necessary here, but
it was done to make clear our intention to have the
b[i]'s as columns in B.
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
c=Table[i+j-1,{i,7},{j,7}];
Unprotect[C];
C=Transpose[c]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1, 2, 3, 4, 5, 6, 7}, {2, 3, 4, 5, 6, 7, 8}, 
 
   {3, 4, 5, 6, 7, 8, 9}, {4, 5, 6, 7, 8, 9, 10}, 
 
   {5, 6, 7, 8, 9, 10, 11}, {6, 7, 8, 9, 10, 11, 12}, 
 
   {7, 8, 9, 10, 11, 12, 13}}]
;[o]
1    2    3    4    5    6    7

2    3    4    5    6    7    8

3    4    5    6    7    8    9

4    5    6    7    8    9    10

5    6    7    8    9    10   11

6    7    8    9    10   11   12

7    8    9    10   11   12   13
:[font = input; initialization; preserveAspect; startGroup]
*)
S=C.Inverse[B]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{11/3, 8/3, 5/3, 2/3, -1/3, -4/3, -7/3}, 
 
   {23/6, 17/6, 11/6, 5/6, -1/6, -7/6, -13/6}, 
 
   {4, 3, 2, 1, 0, -1, -2}, 
 
   {25/6, 19/6, 13/6, 7/6, 1/6, -5/6, -11/6}, 
 
   {13/3, 10/3, 7/3, 4/3, 1/3, -2/3, -5/3}, 
 
   {9/2, 7/2, 5/2, 3/2, 1/2, -1/2, -3/2}, 
 
   {14/3, 11/3, 8/3, 5/3, 2/3, -1/3, -4/3}}]
;[o]
11      8       5       2         1       4       7
--      -       -       -       -(-)    -(-)    -(-)
3       3       3       3         3       3       3

23      17      11      5         1       7       13
--      --      --      -       -(-)    -(-)    -(--)
6       6       6       6         6       6       6



4       3       2       1       0       -1      -2

25      19      13      7       1         5       11
--      --      --      -       -       -(-)    -(--)
6       6       6       6       6         6       6

13      10      7       4       1         2       5
--      --      -       -       -       -(-)    -(-)
3       3       3       3       3         3       3

9       7       5       3       1         1       3
-       -       -       -       -       -(-)    -(-)
2       2       2       2       2         2       2

14      11      8       5       2         1       4
--      --      -       -       -       -(-)    -(-)
3       3       3       3       3         3       3
:[font = input; initialization; preserveAspect]
*)
T[v_]:=S.v
(*
:[font = input; initialization; preserveAspect]
*)
Solution check:
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
Do[Print[T[b[[i]]]-c[[i]]],{i,7}]
(*
:[font = print; inactive; preserveAspect; endGroup]
{0, 0, 0, 0, 0, 0, 0}
{0, 0, 0, 0, 0, 0, 0}
{0, 0, 0, 0, 0, 0, 0}
{0, 0, 0, 0, 0, 0, 0}
{0, 0, 0, 0, 0, 0, 0}
{0, 0, 0, 0, 0, 0, 0}
{0, 0, 0, 0, 0, 0, 0}
:[font = input; initialization; preserveAspect]
*)
Exercise 4.2.3
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
NullSpace[S]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{5, -6, 0, 0, 0, 0, 1}, {4, -5, 0, 0, 0, 1, 0}, 
 
   {3, -4, 0, 0, 1, 0, 0}, {2, -3, 0, 1, 0, 0, 0}, 
 
   {1, -2, 1, 0, 0, 0, 0}}]
;[o]
5    -6   0    0    0    0    1

4    -5   0    0    0    1    0

3    -4   0    0    1    0    0

2    -3   0    1    0    0    0

1    -2   1    0    0    0    0
:[font = input; initialization; preserveAspect; startGroup]
*)
RowReduce[Transpose[S]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1, 0, -1, -2, -3, -4, -5}, 
 
   {0, 1, 2, 3, 4, 5, 6}, {0, 0, 0, 0, 0, 0, 0}, 
 
   {0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0}, 
 
   {0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0}}]
;[o]
1    0    -1   -2   -3   -4   -5

0    1    2    3    4    5    6

0    0    0    0    0    0    0

0    0    0    0    0    0    0

0    0    0    0    0    0    0

0    0    0    0    0    0    0

0    0    0    0    0    0    0
:[font = input; initialization; preserveAspect; startGroup]
*)
{%[[1]],%[[2]]}
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1, 0, -1, -2, -3, -4, -5}, 
 
   {0, 1, 2, 3, 4, 5, 6}}]
;[o]
1    0    -1   -2   -3   -4   -5

0    1    2    3    4    5    6
:[font = input; initialization; preserveAspect]
*)
Exercise 4.2.4
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
B=Array[Min,{6,6}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1, 1, 1, 1, 1, 1}, {1, 2, 2, 2, 2, 2}, 
 
   {1, 2, 3, 3, 3, 3}, {1, 2, 3, 4, 4, 4}, 
 
   {1, 2, 3, 4, 5, 5}, {1, 2, 3, 4, 5, 6}}]
;[o]
1   1   1   1   1   1

1   2   2   2   2   2

1   2   3   3   3   3

1   2   3   4   4   4

1   2   3   4   5   5

1   2   3   4   5   6
:[font = input; initialization; preserveAspect]
*)
Crd[v_]:=Inverse[B].v
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
u=Array[U,6]
v=Array[V,6]
(*
:[font = output; output; inactive; preserveAspect]
{U[1], U[2], U[3], U[4], U[5], U[6]}
;[o]
{U[1], U[2], U[3], U[4], U[5], U[6]}
:[font = output; output; inactive; preserveAspect; endGroup]
{V[1], V[2], V[3], V[4], V[5], V[6]}
;[o]
{V[1], V[2], V[3], V[4], V[5], V[6]}
:[font = input; initialization; preserveAspect]
*)
a)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
Crd[u+v]-(Crd[u]+Crd[v])//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]
*)
Crd[k u]-k Crd[u]//Simplify
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{0, 0, 0, 0, 0, 0}
;[o]
{0, 0, 0, 0, 0, 0}
:[font = input; initialization; preserveAspect]
*)
Exercise 4.2.5
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
S=Inverse[B]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{2, -1, 0, 0, 0, 0}, {-1, 2, -1, 0, 0, 0}, 
 
   {0, -1, 2, -1, 0, 0}, {0, 0, -1, 2, -1, 0}, 
 
   {0, 0, 0, -1, 2, -1}, {0, 0, 0, 0, -1, 1}}]
;[o]
2    -1   0    0    0    0

-1   2    -1   0    0    0

0    -1   2    -1   0    0

0    0    -1   2    -1   0

0    0    0    -1   2    -1

0    0    0    0    -1   1
:[font = input; initialization; preserveAspect]
*)
Exercise 4.2.6
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
f=Table[Min[j-i+1,1],{i,6},{j,9}];
F=Transpose[f]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1, 0, -1, -2, -3, -4}, {1, 1, 0, -1, -2, -3}, 
 
   {1, 1, 1, 0, -1, -2}, {1, 1, 1, 1, 0, -1}, 
 
   {1, 1, 1, 1, 1, 0}, {1, 1, 1, 1, 1, 1}, 
 
   {1, 1, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 1}, 
 
   {1, 1, 1, 1, 1, 1}}]
;[o]
1    0    -1   -2   -3   -4

1    1    0    -1   -2   -3

1    1    1    0    -1   -2

1    1    1    1    0    -1

1    1    1    1    1    0

1    1    1    1    1    1

1    1    1    1    1    1

1    1    1    1    1    1

1    1    1    1    1    1
:[font = input; initialization; preserveAspect; startGroup]
*)
d=Table[1+Min[i,j],{i,6},{j,9}];
Unprotect[D];
D=Transpose[d]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{2, 2, 2, 2, 2, 2}, {2, 3, 3, 3, 3, 3}, 
 
   {2, 3, 4, 4, 4, 4}, {2, 3, 4, 5, 5, 5}, 
 
   {2, 3, 4, 5, 6, 6}, {2, 3, 4, 5, 6, 7}, 
 
   {2, 3, 4, 5, 6, 7}, {2, 3, 4, 5, 6, 7}, 
 
   {2, 3, 4, 5, 6, 7}}]
;[o]
2   2   2   2   2   2

2   3   3   3   3   3

2   3   4   4   4   4

2   3   4   5   5   5

2   3   4   5   6   6

2   3   4   5   6   7

2   3   4   5   6   7

2   3   4   5   6   7

2   3   4   5   6   7
:[font = input; initialization; preserveAspect; startGroup]
*)
SL=Array[LinearSolve[B,D[[#]]]&,9]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{2, 0, 0, 0, 0, 0}, {1, 1, 0, 0, 0, 0}, 
 
   {1, 0, 1, 0, 0, 0}, {1, 0, 0, 1, 0, 0}, 
 
   {1, 0, 0, 0, 1, 0}, {1, 0, 0, 0, 0, 1}, 
 
   {1, 0, 0, 0, 0, 1}, {1, 0, 0, 0, 0, 1}, 
 
   {1, 0, 0, 0, 0, 1}}]
;[o]
2   0   0   0   0   0

1   1   0   0   0   0

1   0   1   0   0   0

1   0   0   1   0   0

1   0   0   0   1   0

1   0   0   0   0   1

1   0   0   0   0   1

1   0   0   0   0   1

1   0   0   0   0   1
:[font = input; initialization; preserveAspect]
*)
L[v_]:=SL.v
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
Do[Print[L[B[[i]]]-d[[i]]],{i,6}]
(*
:[font = print; inactive; preserveAspect; endGroup]
{0, 0, 0, 0, 0, 0, 0, 0, 0}
{0, 0, 0, 0, 0, 0, 0, 0, 0}
{0, 0, 0, 0, 0, 0, 0, 0, 0}
{0, 0, 0, 0, 0, 0, 0, 0, 0}
{0, 0, 0, 0, 0, 0, 0, 0, 0}
{0, 0, 0, 0, 0, 0, 0, 0, 0}
:[font = input; initialization; preserveAspect]
*)
Clear[T]
T[vB_]:=LinearSolve[F,SL.B.vB]
(*
:[font = input; initialization; preserveAspect]
*)
Solution check:
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
T[{1,1,1,1,1,1}]-Sum[LinearSolve[F,d[[i]]],{i,6}]
(*
:[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[u+v]-(T[u]+T[v])//Expand
(*
:[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 u]-k T[u]//Expand
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{0, 0, 0, 0, 0, 0}
;[o]
{0, 0, 0, 0, 0, 0}
:[font = input; initialization; preserveAspect]
*)
Exercise 4.2.7
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
ST=Array[LinearSolve[f,IdentityMatrix[6][[#]]]&,6].SL.B
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{2, 2, 3, 4, 5, 6}, {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]
2   2   3   4   5   6

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]
*)
Solution check:
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
T[u]-ST.u//Simplify
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{0, 0, 0, 0, 0, 0}
;[o]
{0, 0, 0, 0, 0, 0}
:[font = input; initialization; preserveAspect]
*)
Exercise 4.2.8
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
A=Table[i+j,{i,5},{j,7}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{2, 3, 4, 5, 6, 7, 8}, {3, 4, 5, 6, 7, 8, 9}, 
 
   {4, 5, 6, 7, 8, 9, 10}, {5, 6, 7, 8, 9, 10, 11}, 
 
   {6, 7, 8, 9, 10, 11, 12}}]
;[o]
2    3    4    5    6    7    8

3    4    5    6    7    8    9

4    5    6    7    8    9    10

5    6    7    8    9    10   11

6    7    8    9    10   11   12
:[font = input; initialization; preserveAspect; startGroup]
*)
b={35,42,49,56,63}
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{35, 42, 49, 56, 63}
;[o]
{35, 42, 49, 56, 63}
:[font = input; initialization; preserveAspect; startGroup]
*)
sp=LinearSolve[A,b]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{-14, 21, 0, 0, 0, 0, 0}
;[o]
{-14, 21, 0, 0, 0, 0, 0}
:[font = input; initialization; preserveAspect; startGroup]
*)
NullSpace[A]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{5, -6, 0, 0, 0, 0, 1}, {4, -5, 0, 0, 0, 1, 0}, 
 
   {3, -4, 0, 0, 1, 0, 0}, {2, -3, 0, 1, 0, 0, 0}, 
 
   {1, -2, 1, 0, 0, 0, 0}}]
;[o]
5    -6   0    0    0    0    1

4    -5   0    0    0    1    0

3    -4   0    0    1    0    0

2    -3   0    1    0    0    0

1    -2   1    0    0    0    0
:[font = input; initialization; preserveAspect; startGroup]
*)
sh=Sum[t[i]%[[i]],{i,5}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{5*t[1] + 4*t[2] + 3*t[3] + 2*t[4] + t[5], 
 
  -6*t[1] - 5*t[2] - 4*t[3] - 3*t[4] - 2*t[5], t[5], t[4], 
 
  t[3], t[2], t[1]}
;[o]
{5 t[1] + 4 t[2] + 3 t[3] + 2 t[4] + t[5], 
 
  -6 t[1] - 5 t[2] - 4 t[3] - 3 t[4] - 2 t[5], t[5], t[4], 
 
  t[3], t[2], t[1]}
:[font = input; initialization; preserveAspect; startGroup]
*)
sg=sp+sh
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{-14 + 5*t[1] + 4*t[2] + 3*t[3] + 2*t[4] + t[5], 
 
  21 - 6*t[1] - 5*t[2] - 4*t[3] - 3*t[4] - 2*t[5], t[5], 
 
  t[4], t[3], t[2], t[1]}
;[o]
{-14 + 5 t[1] + 4 t[2] + 3 t[3] + 2 t[4] + t[5], 
 
  21 - 6 t[1] - 5 t[2] - 4 t[3] - 3 t[4] - 2 t[5], t[5], 
 
  t[4], t[3], t[2], t[1]}
:[font = input; initialization; preserveAspect]
*)
Clear[L]
L[x_]:=A.x
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
L[sg]-b//Expand
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{0, 0, 0, 0, 0}
;[o]
{0, 0, 0, 0, 0}
^*)