(*^
::[	Information =

	"This is a Mathematica Notebook file.  It contains ASCII text, and can be
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:[font = input; initialization; preserveAspect]
*)
$Post:=If[MatrixQ[#],MatrixForm[#],#]&
(*
:[font = input; initialization; preserveAspect]
*)
Exercise 2.3.1
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
b=Array[#&,5]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{1, 2, 3, 4, 5}
;[o]
{1, 2, 3, 4, 5}
:[font = input; initialization; preserveAspect]
*)
a)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
Do[v[i]=Table[Mod[6-i-j,5]+1,{j,5}],{i,3}];
{v[1],v[2],v[3]}
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{5, 4, 3, 2, 1}, {4, 3, 2, 1, 5}, 
 
   {3, 2, 1, 5, 4}}]
;[o]
5   4   3   2   1

4   3   2   1   5

3   2   1   5   4
:[font = input; initialization; preserveAspect; startGroup]
*)
LinearSolve[Transpose[Array[v[#]&,3]],b]
(*
:[font = message; inactive; preserveAspect]
LinearSolve::nosol: 
   Linear equation encountered which has no solution.
:[font = output; output; inactive; preserveAspect; endGroup]
LinearSolve[{{5, 4, 3}, {4, 3, 2}, {3, 2, 1}, {2, 1, 5}, 
 
   {1, 5, 4}}, {1, 2, 3, 4, 5}]
;[o]
LinearSolve[{{5, 4, 3}, {4, 3, 2}, {3, 2, 1}, {2, 1, 5}, 
 
   {1, 5, 4}}, {1, 2, 3, 4, 5}]
:[font = input; initialization; preserveAspect]
*)
No solution means b does not lie in the span of v1,v2,v3
(*
:[font = input; initialization; preserveAspect]
*)
b)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
LinearSolve[Transpose[{{2,3,4,5,6},{3,4,5,6,7}}],b]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{2, -1}
;[o]
{2, -1}
:[font = input; initialization; preserveAspect]
*)
We conclude that b is in the span of the two vectors
above.
(*
:[font = input; initialization; preserveAspect]
*)
Exercise 2.3.2
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
b=Array[1&,5]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{1, 1, 1, 1, 1}
;[o]
{1, 1, 1, 1, 1}
:[font = input; initialization; preserveAspect]
*)
a)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
v[1]={5,4,3,2,1}; v[2]={1,2,3,4,5}; v[3]={3,2,1,5,4};
LinearSolve[Transpose[{v[1],v[2],v[3]}],b]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{1/6, 1/6, 0}
;[o]
 1  1
{-, -, 0}
 6  6
:[font = input; initialization; preserveAspect]
*)
b in the span.
(*
:[font = input; initialization; preserveAspect]
*)
b)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
v[1]={2,3,4,5,6}; v[2]={8,7,6,5,4}; v[3]={1,0,1,0,1};
LinearSolve[Transpose[Array[v[#]&,3]],b]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{1/10, 1/10, 0}
;[o]
 1   1
{--, --, 0}
 10  10
:[font = input; initialization; preserveAspect]
*)
b in the span.
(*
:[font = input; initialization; preserveAspect]
*)
Exercise 3.2.3
(*
:[font = input; initialization; preserveAspect]
*)
Do[w[i]=Table[i-j,{j,15}],{i,15}]
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
v=Array[#&,15]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}
;[o]
{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}
:[font = input; initialization; preserveAspect; startGroup]
*)
LinearSolve[Transpose[Array[w[#]&,15]],v]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{-2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}
;[o]
{-2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}
:[font = input; initialization; preserveAspect]
*)
v is in the span of the w's.
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
v=Array[#^2&,15]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 
 
  225}
;[o]
{1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 
 
  225}
:[font = input; initialization; preserveAspect; startGroup]
*)
LinearSolve[Transpose[Array[w[#]&,15]],v]
(*
:[font = message; inactive; preserveAspect]
LinearSolve::nosol: 
   Linear equation encountered which has no solution.
:[font = output; output; inactive; preserveAspect; endGroup]
LinearSolve[{{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 
 
    14}, {-1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}, 
 
   {-2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, 
 
   {-3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}, 
 
   {-4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, 
 
   {-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, 
 
   {-6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8}, 
 
   {-7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7}, 
 
   {-8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6}, 
 
   {-9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5}, 
 
   {-10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4}, 
 
   {-11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 
 
    3}, {-12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 
 
    0, 1, 2}, {-13, -12, -11, -10, -9, -8, -7, -6, -5, -4, 
 
    -3, -2, -1, 0, 1}, {-14, -13, -12, -11, -10, -9, -8, -7, 
 
    -6, -5, -4, -3, -2, -1, 0}}, 
 
  {1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 
 
   225}]
;[o]
LinearSolve[{{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 
 
    14}, {-1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}, 
 
   {-2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, 
 
   {-3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}, 
 
   {-4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, 
 
   {-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, 
 
   {-6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8}, 
 
   {-7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7}, 
 
   {-8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6}, 
 
   {-9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5}, 
 
   {-10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4}, 
 
   {-11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 
 
    3}, {-12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 
 
    0, 1, 2}, {-13, -12, -11, -10, -9, -8, -7, -6, -5, -4, 
 
    -3, -2, -1, 0, 1}, {-14, -13, -12, -11, -10, -9, -8, -7, 
 
    -6, -5, -4, -3, -2, -1, 0}}, 
 
  {1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 
 
   225}]
:[font = input; initialization; preserveAspect]
*)
This v is not in the span.
(*
:[font = input; initialization; preserveAspect]
*)
Clear[v]
Do[v[i]=Table[1/(1+Mod[i-j,5]),{j,10}],{i,10}]
(*
:[font = input; initialization; preserveAspect]
*)
Exercise 2.3.4
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
Det[Array[v[#]&,10]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
0
;[o]
0
:[font = input; initialization; preserveAspect]
*)
The v's are NOT linearly independent.
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
Do[                                                     \
   x=LinearSolve[Transpose[Array[v[#]&,10-i]],v[11-i]]; \
   Print[11-i,": ",x],                                  \
   {i,9}]
(*
:[font = print; inactive; preserveAspect]
10: {0, 0, 0, 0, 1, 0, 0, 0, 0}
9: {0, 0, 0, 1, 0, 0, 0, 0}
8: {0, 0, 1, 0, 0, 0, 0}
7: {0, 1, 0, 0, 0, 0}
6: {1, 0, 0, 0, 0}
                    1  1  1    1     1  1    1  1     1
5: LinearSolve[{{1, -, -, -}, {-, 1, -, -}, {-, -, 1, -}, 
                    2  3  4    5     2  3    4  5     2
 
     1  1  1       1  1  1  1       1  1  1    1     1  1
    {-, -, -, 1}, {-, -, -, -}, {1, -, -, -}, {-, 1, -, -}, 
     3  4  5       2  3  4  5       2  3  4    5     2  3
 
     1  1     1    1  1  1       1  1  1  1
    {-, -, 1, -}, {-, -, -, 1}, {-, -, -, -}}, 
     4  5     2    3  4  5       2  3  4  5
 
    1  1  1  1     1  1  1  1
   {-, -, -, -, 1, -, -, -, -, 1}]
    5  4  3  2     5  4  3  2
                    1  1    1     1    1  1       1  1  1
4: LinearSolve[{{1, -, -}, {-, 1, -}, {-, -, 1}, {-, -, -}, 
                    2  3    5     2    4  5       3  4  5
 
     1  1  1       1  1    1     1    1  1       1  1  1
    {-, -, -}, {1, -, -}, {-, 1, -}, {-, -, 1}, {-, -, -}, 
     2  3  4       2  3    5     2    4  5       3  4  5
 
     1  1  1     1  1  1     1  1  1  1     1
    {-, -, -}}, {-, -, -, 1, -, -, -, -, 1, -}]
     2  3  4     4  3  2     5  4  3  2     5
                    1    1       1  1    1  1    1  1
3: LinearSolve[{{1, -}, {-, 1}, {-, -}, {-, -}, {-, -}, 
                    2    5       4  5    3  4    2  3
 
        1    1       1  1    1  1    1  1
    {1, -}, {-, 1}, {-, -}, {-, -}, {-, -}}, 
        2    5       4  5    3  4    2  3
 
    1  1     1  1  1  1     1  1
   {-, -, 1, -, -, -, -, 1, -, -}]
    3  2     5  4  3  2     5  4
                      1    1    1    1         1    1    1
2: LinearSolve[{{1}, {-}, {-}, {-}, {-}, {1}, {-}, {-}, {-}, 
                      5    4    3    2         5    4    3
 
     1     1     1  1  1  1     1  1  1
    {-}}, {-, 1, -, -, -, -, 1, -, -, -}]
     2     2     5  4  3  2     5  4  3
:[font = message; inactive; preserveAspect]
LinearSolve::nosol: 
   Linear equation encountered which has no solution.
:[font = message; inactive; preserveAspect]
LinearSolve::nosol: 
   Linear equation encountered which has no solution.
:[font = message; inactive; preserveAspect]
LinearSolve::nosol: 
   Linear equation encountered which has no solution.
:[font = message; inactive; preserveAspect; endGroup]
General::stop: 
   Further output of LinearSolve::nosol
     will be suppressed during this calculation.
:[font = input; initialization; preserveAspect]
*)
Vectors 6-10 are linear combinations of the preceding,
2-5 are not.
(*
:[font = input; initialization; preserveAspect]
*)
Exercise 2.3.6
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
Length[NullSpace[Transpose[Array[v[#]&,5]]]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
0
;[o]
0
:[font = input; initialization; preserveAspect]
*)
The vectors 1-5 ARE linearly independent.
(*
:[font = input; initialization; preserveAspect]
*)
Exercise 2.3.7
(*
:[font = input; initialization; preserveAspect]
*)
See exercise 2.3.5

(*
^*)