(*^
::[	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]
*)
$Post:=If[MatrixQ[#],TableForm[#],#]&
(*
:[font = input; initialization; preserveAspect]
*)
Exercise 2.5.1
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
A=Array[Random[Integer,{-100,100}]&,{5,7}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
TableForm[{{72, 90, 84, -99, 51, 46, 51}, 
 
   {20, 25, -85, 47, 51, 3, 25}, 
 
   {-27, 4, -52, 68, 13, 71, 89}, 
 
   {-100, -44, 49, -70, 8, 61, 47}, 
 
   {13, -86, -64, 76, -31, 17, 77}}]
;[o]
72     90    84    -99   51    46   51

20     25    -85   47    51    3    25

-27    4     -52   68    13    71   89

-100   -44   49    -70   8     61   47

13     -86   -64   76    -31   17   77
:[font = input; initialization; preserveAspect; startGroup]
*)
RowReduce[A]//TableForm
(*
:[font = output; output; inactive; preserveAspect; endGroup]
TableForm[{{1, 0, 0, 0, 0, -79178591/1033744074, 
 
    144829265/344581358}, 
 
   {0, 1, 0, 0, 0, -763957988/516872037, 
 
    -430490341/172290679}, 
 
   {0, 0, 1, 0, 0, 1784645774/516872037, 
 
    769297328/172290679}, 
 
   {0, 0, 0, 1, 0, 1039560391/344581358, 
 
    1387881911/344581358}, 
 
   {0, 0, 0, 0, 1, 3915577841/1033744074, 
 
    1819461993/344581358}}]
;[o]
                       79178591     144829265
                    -(----------)   ---------
1   0   0   0   0     1033744074    344581358

                      763957988       430490341
                    -(---------)    -(---------)
0   1   0   0   0     516872037       172290679

                    1784645774      769297328
                    ----------      ---------
0   0   1   0   0   516872037       172290679

                    1039560391      1387881911
                    ----------      ----------
0   0   0   1   0   344581358       344581358

                    3915577841      1819461993
                    ----------      ----------
0   0   0   0   1   1033744074      344581358
:[font = input; initialization; preserveAspect; startGroup]
*)
RowReduce[Transpose[A]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
TableForm[{{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}, 
 
   {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}}]
;[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

0   0   0   0   0

0   0   0   0   0
:[font = input; initialization; preserveAspect]
*)
By inspection, one sees that the row rank above is
equal to the column rank (and therefore the dimensions
of row space and column space are the same).
(*
:[font = input; initialization; preserveAspect]
*)
Exercise 2.5.2
(*
:[font = input; initialization; preserveAspect]
*)
f[i_,j_]:=i+j
(*
:[font = input; initialization; preserveAspect]
*)
a)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
A=Array[f,{5,13}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
TableForm[{{2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14}, 
 
   {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}, 
 
   {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16}, 
 
   {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17}, 
 
   {6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18}}]
;[o]
2   3   4   5   6    7    8    9    10   11   12   13   14

3   4   5   6   7    8    9    10   11   12   13   14   15

4   5   6   7   8    9    10   11   12   13   14   15   16

5   6   7   8   9    10   11   12   13   14   15   16   17

6   7   8   9   10   11   12   13   14   15   16   17   18
:[font = input; initialization; preserveAspect; startGroup]
*)
NullSpace[A]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
TableForm[{{11, -12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1}, 
 
   {10, -11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0}, 
 
   {9, -10, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0}, 
 
   {8, -9, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0}, 
 
   {7, -8, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0}, 
 
   {6, -7, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0}, 
 
   {5, -6, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0}, 
 
   {4, -5, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0}, 
 
   {3, -4, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0}, 
 
   {2, -3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0}, 
 
   {1, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}}]
;[o]
11   -12   0   0   0   0   0   0   0   0   0   0   1

10   -11   0   0   0   0   0   0   0   0   0   1   0

9    -10   0   0   0   0   0   0   0   0   1   0   0

8    -9    0   0   0   0   0   0   0   1   0   0   0

7    -8    0   0   0   0   0   0   1   0   0   0   0

6    -7    0   0   0   0   0   1   0   0   0   0   0

5    -6    0   0   0   0   1   0   0   0   0   0   0

4    -5    0   0   0   1   0   0   0   0   0   0   0

3    -4    0   0   1   0   0   0   0   0   0   0   0

2    -3    0   1   0   0   0   0   0   0   0   0   0

1    -2    1   0   0   0   0   0   0   0   0   0   0
:[font = input; initialization; preserveAspect; startGroup]
*)
RowReduce[A]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
TableForm[{{1, 0, -1, -2, -3, -4, -5, -6, -7, -8, -9, -10, 
 
    -11}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, 
 
   {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}}]
;[o]
1   0   -1   -2   -3   -4   -5   -6   -7   -8   -9   -10
 
      -11

0   1   2    3    4    5    6    7    8    9    10   11
 
      12

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; startGroup]
*)
RowSpaceBasis={%[[1]],%[[2]]}
(*
:[font = output; output; inactive; preserveAspect; endGroup]
TableForm[{{1, 0, -1, -2, -3, -4, -5, -6, -7, -8, -9, -10, 
 
    -11}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}}]
;[o]
1   0   -1   -2   -3   -4   -5   -6   -7   -8   -9   -10
 
      -11

0   1   2    3    4    5    6    7    8    9    10   11
 
      12
:[font = input; initialization; preserveAspect; startGroup]
*)
RowReduce[Transpose[A]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
TableForm[{{1, 0, -1, -2, -3}, {0, 1, 2, 3, 4}, 
 
   {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, 0}}]
;[o]
1   0   -1   -2   -3

0   1   2    3    4

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    0
:[font = input; initialization; preserveAspect; startGroup]
*)
ColumnSpaceBasis={%[[1]],%[[2]]}
(*
:[font = output; output; inactive; preserveAspect; endGroup]
TableForm[{{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]
*)
A=Array[f,{11,7}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
TableForm[{{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}, 
 
   {8, 9, 10, 11, 12, 13, 14}, {9, 10, 11, 12, 13, 14, 15}, 
 
   {10, 11, 12, 13, 14, 15, 16}, 
 
   {11, 12, 13, 14, 15, 16, 17}, 
 
   {12, 13, 14, 15, 16, 17, 18}}]
;[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

7    8    9    10   11   12   13

8    9    10   11   12   13   14

9    10   11   12   13   14   15

10   11   12   13   14   15   16

11   12   13   14   15   16   17

12   13   14   15   16   17   18
:[font = input; initialization; preserveAspect; startGroup]
*)
NullSpace[A]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
TableForm[{{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[A]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
TableForm[{{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}, 
 
   {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

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]
*)
RowSpaceBasis={%[[1]],%[[2]]}
(*
:[font = output; output; inactive; preserveAspect; endGroup]
TableForm[{{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; startGroup]
*)
RowReduce[Transpose[A]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
TableForm[{{1, 0, -1, -2, -3, -4, -5, -6, -7, -8, -9}, 
 
   {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, 
 
   {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, 0}}]
;[o]
1   0   -1   -2   -3   -4   -5   -6   -7   -8   -9

0   1   2    3    4    5    6    7    8    9    10

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    0
:[font = input; initialization; preserveAspect; startGroup]
*)
ColumnSpaceBasis={%[[1]],%[[2]]}
(*
:[font = output; output; inactive; preserveAspect; endGroup]
TableForm[{{1, 0, -1, -2, -3, -4, -5, -6, -7, -8, -9}, 
 
   {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}}]
;[o]
1   0   -1   -2   -3   -4   -5   -6   -7   -8   -9

0   1   2    3    4    5    6    7    8    9    10
:[font = input; initialization; preserveAspect]
*)
c)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
A=Array[f,{16,8}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
TableForm[{{2, 3, 4, 5, 6, 7, 8, 9}, 
 
   {3, 4, 5, 6, 7, 8, 9, 10}, {4, 5, 6, 7, 8, 9, 10, 11}, 
 
   {5, 6, 7, 8, 9, 10, 11, 12}, 
 
   {6, 7, 8, 9, 10, 11, 12, 13}, 
 
   {7, 8, 9, 10, 11, 12, 13, 14}, 
 
   {8, 9, 10, 11, 12, 13, 14, 15}, 
 
   {9, 10, 11, 12, 13, 14, 15, 16}, 
 
   {10, 11, 12, 13, 14, 15, 16, 17}, 
 
   {11, 12, 13, 14, 15, 16, 17, 18}, 
 
   {12, 13, 14, 15, 16, 17, 18, 19}, 
 
   {13, 14, 15, 16, 17, 18, 19, 20}, 
 
   {14, 15, 16, 17, 18, 19, 20, 21}, 
 
   {15, 16, 17, 18, 19, 20, 21, 22}, 
 
   {16, 17, 18, 19, 20, 21, 22, 23}, 
 
   {17, 18, 19, 20, 21, 22, 23, 24}}]
;[o]
2    3    4    5    6    7    8    9

3    4    5    6    7    8    9    10

4    5    6    7    8    9    10   11

5    6    7    8    9    10   11   12

6    7    8    9    10   11   12   13

7    8    9    10   11   12   13   14

8    9    10   11   12   13   14   15

9    10   11   12   13   14   15   16

10   11   12   13   14   15   16   17

11   12   13   14   15   16   17   18

12   13   14   15   16   17   18   19

13   14   15   16   17   18   19   20

14   15   16   17   18   19   20   21

15   16   17   18   19   20   21   22

16   17   18   19   20   21   22   23

17   18   19   20   21   22   23   24
:[font = input; initialization; preserveAspect; startGroup]
*)
NullSpace[A]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
TableForm[{{6, -7, 0, 0, 0, 0, 0, 1}, 
 
   {5, -6, 0, 0, 0, 0, 1, 0}, {4, -5, 0, 0, 0, 1, 0, 0}, 
 
   {3, -4, 0, 0, 1, 0, 0, 0}, {2, -3, 0, 1, 0, 0, 0, 0}, 
 
   {1, -2, 1, 0, 0, 0, 0, 0}}]
;[o]
6   -7   0   0   0   0   0   1

5   -6   0   0   0   0   1   0

4   -5   0   0   0   1   0   0

3   -4   0   0   1   0   0   0

2   -3   0   1   0   0   0   0

1   -2   1   0   0   0   0   0
:[font = input; initialization; preserveAspect; startGroup]
*)
RowReduce[A]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
TableForm[{{1, 0, -1, -2, -3, -4, -5, -6}, 
 
   {0, 1, 2, 3, 4, 5, 6, 7}, {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, 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, 0, 0, 0, 0}}]
;[o]
1   0   -1   -2   -3   -4   -5   -6

0   1   2    3    4    5    6    7

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    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    0    0    0    0
:[font = input; initialization; preserveAspect; startGroup]
*)
RowSpaceBasis={%[[1]],%[[2]]}
(*
:[font = output; output; inactive; preserveAspect; endGroup]
TableForm[{{1, 0, -1, -2, -3, -4, -5, -6}, 
 
   {0, 1, 2, 3, 4, 5, 6, 7}}]
;[o]
1   0   -1   -2   -3   -4   -5   -6

0   1   2    3    4    5    6    7
:[font = input; initialization; preserveAspect; startGroup]
*)
RowReduce[Transpose[A]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
TableForm[{{1, 0, -1, -2, -3, -4, -5, -6, -7, -8, -9, -10, 
 
    -11, -12, -13, -14}, 
 
   {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}, 
 
   {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, 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}}]
;[o]
1   0   -1   -2   -3   -4   -5   -6   -7   -8   -9   -10
 
      -11   -12   -13   -14

0   1   2    3    4    5    6    7    8    9    10   11
 
      12    13    14    15

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    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; startGroup]
*)
ColumnSpaceBasis={%[[1]],%[[2]]}
(*
:[font = output; output; inactive; preserveAspect; endGroup]
TableForm[{{1, 0, -1, -2, -3, -4, -5, -6, -7, -8, -9, -10, 
 
    -11, -12, -13, -14}, 
 
   {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}}]
;[o]
1   0   -1   -2   -3   -4   -5   -6   -7   -8   -9   -10
 
      -11   -12   -13   -14

0   1   2    3    4    5    6    7    8    9    10   11
 
      12    13    14    15
:[font = input; initialization; preserveAspect]
*)
Exercise 2.5.3
(*
:[font = input; initialization; preserveAspect]
*)
Yes, it generalizes to the fact that the rank of the
matrix ( = dimension of the column space = dimension
of the row space) and the dimension of the null space
sum up to the number of columns. Reason being that each
column will either
            - be linearly independent of the preceding,
              and therefore able to be part of a basis
              for the column space,
            or
            - be a linear combination of the preceding,
              in which case the corresponding x[i] it
              multiplies in Ax=0 is arbitrary, therefore
              it generates one more dimension for the
              null space.
(*
:[font = input; initialization; preserveAspect]
*)
Exercise 2.5.4
(*
:[font = input; initialization; preserveAspect]
*)
h[i_,j_]:=1/(i+j-1)
(*
:[font = input; initialization; preserveAspect]
*)
a)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
A=Array[h,{13,15}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
TableForm[{{1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9, 1/10, 
 
    1/11, 1/12, 1/13, 1/14, 1/15}, 
 
   {1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9, 1/10, 1/11, 
 
    1/12, 1/13, 1/14, 1/15, 1/16}, 
 
   {1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9, 1/10, 1/11, 1/12, 
 
    1/13, 1/14, 1/15, 1/16, 1/17}, 
 
   {1/4, 1/5, 1/6, 1/7, 1/8, 1/9, 1/10, 1/11, 1/12, 1/13, 
 
    1/14, 1/15, 1/16, 1/17, 1/18}, 
 
   {1/5, 1/6, 1/7, 1/8, 1/9, 1/10, 1/11, 1/12, 1/13, 1/14, 
 
    1/15, 1/16, 1/17, 1/18, 1/19}, 
 
   {1/6, 1/7, 1/8, 1/9, 1/10, 1/11, 1/12, 1/13, 1/14, 1/15, 
 
    1/16, 1/17, 1/18, 1/19, 1/20}, 
 
   {1/7, 1/8, 1/9, 1/10, 1/11, 1/12, 1/13, 1/14, 1/15, 1/16, 
 
    1/17, 1/18, 1/19, 1/20, 1/21}, 
 
   {1/8, 1/9, 1/10, 1/11, 1/12, 1/13, 1/14, 1/15, 1/16, 
 
    1/17, 1/18, 1/19, 1/20, 1/21, 1/22}, 
 
   {1/9, 1/10, 1/11, 1/12, 1/13, 1/14, 1/15, 1/16, 1/17, 
 
    1/18, 1/19, 1/20, 1/21, 1/22, 1/23}, 
 
   {1/10, 1/11, 1/12, 1/13, 1/14, 1/15, 1/16, 1/17, 1/18, 
 
    1/19, 1/20, 1/21, 1/22, 1/23, 1/24}, 
 
   {1/11, 1/12, 1/13, 1/14, 1/15, 1/16, 1/17, 1/18, 1/19, 
 
    1/20, 1/21, 1/22, 1/23, 1/24, 1/25}, 
 
   {1/12, 1/13, 1/14, 1/15, 1/16, 1/17, 1/18, 1/19, 1/20, 
 
    1/21, 1/22, 1/23, 1/24, 1/25, 1/26}, 
 
   {1/13, 1/14, 1/15, 1/16, 1/17, 1/18, 1/19, 1/20, 1/21, 
 
    1/22, 1/23, 1/24, 1/25, 1/26, 1/27}}]
;[o]
     1    1    1    1    1    1    1    1    1    1    1
     -    -    -    -    -    -    -    -    --   --   --
1    2    3    4    5    6    7    8    9    10   11   12
 
      1    1    1
      --   --   --
      13   14   15

1    1    1    1    1    1    1    1    1    1    1    1
-    -    -    -    -    -    -    -    --   --   --   --
2    3    4    5    6    7    8    9    10   11   12   13
 
      1    1    1
      --   --   --
      14   15   16

1    1    1    1    1    1    1    1    1    1    1    1
-    -    -    -    -    -    -    --   --   --   --   --
3    4    5    6    7    8    9    10   11   12   13   14
 
      1    1    1
      --   --   --
      15   16   17

1    1    1    1    1    1    1    1    1    1    1    1
-    -    -    -    -    -    --   --   --   --   --   --
4    5    6    7    8    9    10   11   12   13   14   15
 
      1    1    1
      --   --   --
      16   17   18

1    1    1    1    1    1    1    1    1    1    1    1
-    -    -    -    -    --   --   --   --   --   --   --
5    6    7    8    9    10   11   12   13   14   15   16
 
      1    1    1
      --   --   --
      17   18   19

1    1    1    1    1    1    1    1    1    1    1    1
-    -    -    -    --   --   --   --   --   --   --   --
6    7    8    9    10   11   12   13   14   15   16   17
 
      1    1    1
      --   --   --
      18   19   20

1    1    1    1    1    1    1    1    1    1    1    1
-    -    -    --   --   --   --   --   --   --   --   --
7    8    9    10   11   12   13   14   15   16   17   18
 
      1    1    1
      --   --   --
      19   20   21

1    1    1    1    1    1    1    1    1    1    1    1
-    -    --   --   --   --   --   --   --   --   --   --
8    9    10   11   12   13   14   15   16   17   18   19
 
      1    1    1
      --   --   --
      20   21   22

1    1    1    1    1    1    1    1    1    1    1    1
-    --   --   --   --   --   --   --   --   --   --   --
9    10   11   12   13   14   15   16   17   18   19   20
 
      1    1    1
      --   --   --
      21   22   23

1    1    1    1    1    1    1    1    1    1    1    1
--   --   --   --   --   --   --   --   --   --   --   --
10   11   12   13   14   15   16   17   18   19   20   21
 
      1    1    1
      --   --   --
      22   23   24

1    1    1    1    1    1    1    1    1    1    1    1
--   --   --   --   --   --   --   --   --   --   --   --
11   12   13   14   15   16   17   18   19   20   21   22
 
      1    1    1
      --   --   --
      23   24   25

1    1    1    1    1    1    1    1    1    1    1    1
--   --   --   --   --   --   --   --   --   --   --   --
12   13   14   15   16   17   18   19   20   21   22   23
 
      1    1    1
      --   --   --
      24   25   26

1    1    1    1    1    1    1    1    1    1    1    1
--   --   --   --   --   --   --   --   --   --   --   --
13   14   15   16   17   18   19   20   21   22   23   24
 
      1    1    1
      --   --   --
      25   26   27
:[font = input; initialization; preserveAspect; startGroup]
*)
NullSpace[A]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
TableForm[{{-13/20058300, 196/1671525, -7007/1337220, 
 
    20384/200583, -7007/6555, 224224/32775, -49049/1725, 
 
    9152/115, -7007/46, 616616/3105, -77077/450, 20384/225, 
 
    -637/27, 0, 1}, {-1/10400600, 13/742900, -117/148580, 
 
    572/37145, -143/874, 11583/10925, -2574/575, 10296/805, 
 
    -11583/460, 1573/46, -1573/50, 468/25, -13/2, 1, 0}}]
;[o]
     13         196        7007      20384      7007
-(--------)   -------   -(-------)   ------   -(----)
  20058300    1671525     1337220    200583     6555
 
   224224     49049    9152      7007     616616
   ------   -(-----)   ----    -(----)    ------
   32775      1725     115        46       3105
 
     77077    20384     637
   -(-----)   -----   -(---)
      450      225      27     0   1

     1          13         117        572       143
-(--------)   ------    -(------)    -----    -(---)
  10400600    742900      148580     37145      874
 
   11583      2574     10296     11583    1573
   -----    -(----)    -----   -(-----)   ----
   10925      575       805       460      46
 
     1573     468       13
   -(----)    ---     -(--)
      50      25        2      1   0
:[font = input; initialization; preserveAspect; startGroup]
*)
RowSpaceBasis=RowReduce[A]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
TableForm[{{1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
 
    1/10400600, 13/20058300}, 
 
   {0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -13/742900, 
 
    -196/1671525}, {0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
 
    117/148580, 7007/1337220}, 
 
   {0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -572/37145, 
 
    -20384/200583}, {0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 
 
    143/874, 7007/6555}, 
 
   {0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, -11583/10925, 
 
    -224224/32775}, {0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 
 
    2574/575, 49049/1725}, 
 
   {0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, -10296/805, 
 
    -9152/115}, {0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 
 
    11583/460, 7007/46}, 
 
   {0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, -1573/46, 
 
    -616616/3105}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 
 
    1573/50, 77077/450}, 
 
   {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -468/25, 
 
    -20384/225}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 
 
    13/2, 637/27}}]
;[o]


1   0   0   0   0   0   0   0   0   0   0   0   0
 
      1           13
   --------    --------
   10400600    20058300



0   1   0   0   0   0   0   0   0   0   0   0   0
 
       13          196
   -(------)   -(-------)
     742900      1671525



0   0   1   0   0   0   0   0   0   0   0   0   0
 
    117         7007
   ------      -------
   148580      1337220



0   0   0   1   0   0   0   0   0   0   0   0   0
 
      572        20384
   -(-----)    -(------)
     37145       200583



0   0   0   0   1   0   0   0   0   0   0   0   0
 
   143         7007
   ---         ----
   874         6555



0   0   0   0   0   1   0   0   0   0   0   0   0
 
     11583       224224
   -(-----)    -(------)
     10925       32775



0   0   0   0   0   0   1   0   0   0   0   0   0
 
   2574        49049
   ----        -----
   575         1725



0   0   0   0   0   0   0   1   0   0   0   0   0
 
     10296       9152
   -(-----)    -(----)
      805        115



0   0   0   0   0   0   0   0   1   0   0   0   0
 
   11583       7007
   -----       ----
    460         46



0   0   0   0   0   0   0   0   0   1   0   0   0
 
     1573        616616
   -(----)     -(------)
      46          3105



0   0   0   0   0   0   0   0   0   0   1   0   0
 
   1573        77077
   ----        -----
    50          450



0   0   0   0   0   0   0   0   0   0   0   1   0
 
     468         20384
   -(---)      -(-----)
     25           225



0   0   0   0   0   0   0   0   0   0   0   0   1
 
   13          637
   --          ---
   2           27
:[font = input; initialization; preserveAspect; startGroup]
*)
RowReduce[Transpose[A]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
TableForm[{{1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, 
 
   {0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, 
 
   {0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, 
 
   {0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0}, 
 
   {0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0}, 
 
   {0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0}, 
 
   {0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0}, 
 
   {0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0}, 
 
   {0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0}, 
 
   {0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0}, 
 
   {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0}, 
 
   {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0}, 
 
   {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1}, 
 
   {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   0   0   0   0   0   0   0   0   0   0   0

0   1   0   0   0   0   0   0   0   0   0   0   0

0   0   1   0   0   0   0   0   0   0   0   0   0

0   0   0   1   0   0   0   0   0   0   0   0   0

0   0   0   0   1   0   0   0   0   0   0   0   0

0   0   0   0   0   1   0   0   0   0   0   0   0

0   0   0   0   0   0   1   0   0   0   0   0   0

0   0   0   0   0   0   0   1   0   0   0   0   0

0   0   0   0   0   0   0   0   1   0   0   0   0

0   0   0   0   0   0   0   0   0   1   0   0   0

0   0   0   0   0   0   0   0   0   0   1   0   0

0   0   0   0   0   0   0   0   0   0   0   1   0

0   0   0   0   0   0   0   0   0   0   0   0   1

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]
*)
ColumnSpaceBasis=Array[%[[#]]&,13]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
TableForm[{{1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, 
 
   {0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, 
 
   {0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, 
 
   {0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0}, 
 
   {0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0}, 
 
   {0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0}, 
 
   {0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0}, 
 
   {0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0}, 
 
   {0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0}, 
 
   {0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0}, 
 
   {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0}, 
 
   {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0}, 
 
   {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1}}]
;[o]
1   0   0   0   0   0   0   0   0   0   0   0   0

0   1   0   0   0   0   0   0   0   0   0   0   0

0   0   1   0   0   0   0   0   0   0   0   0   0

0   0   0   1   0   0   0   0   0   0   0   0   0

0   0   0   0   1   0   0   0   0   0   0   0   0

0   0   0   0   0   1   0   0   0   0   0   0   0

0   0   0   0   0   0   1   0   0   0   0   0   0

0   0   0   0   0   0   0   1   0   0   0   0   0

0   0   0   0   0   0   0   0   1   0   0   0   0

0   0   0   0   0   0   0   0   0   1   0   0   0

0   0   0   0   0   0   0   0   0   0   1   0   0

0   0   0   0   0   0   0   0   0   0   0   1   0

0   0   0   0   0   0   0   0   0   0   0   0   1
:[font = input; initialization; preserveAspect]
*)
b)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
A=Array[h,{14,3}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
TableForm[{{1, 1/2, 1/3}, {1/2, 1/3, 1/4}, {1/3, 1/4, 1/5}, 
 
   {1/4, 1/5, 1/6}, {1/5, 1/6, 1/7}, {1/6, 1/7, 1/8}, 
 
   {1/7, 1/8, 1/9}, {1/8, 1/9, 1/10}, {1/9, 1/10, 1/11}, 
 
   {1/10, 1/11, 1/12}, {1/11, 1/12, 1/13}, 
 
   {1/12, 1/13, 1/14}, {1/13, 1/14, 1/15}, 
 
   {1/14, 1/15, 1/16}}]
;[o]
     1    1
     -    -
1    2    3

1    1    1
-    -    -
2    3    4

1    1    1
-    -    -
3    4    5

1    1    1
-    -    -
4    5    6

1    1    1
-    -    -
5    6    7

1    1    1
-    -    -
6    7    8

1    1    1
-    -    -
7    8    9

1    1    1
-    -    --
8    9    10

1    1    1
-    --   --
9    10   11

1    1    1
--   --   --
10   11   12

1    1    1
--   --   --
11   12   13

1    1    1
--   --   --
12   13   14

1    1    1
--   --   --
13   14   15

1    1    1
--   --   --
14   15   16
:[font = input; initialization; preserveAspect; startGroup]
*)
NullSpace[A]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{}
;[o]
{}
:[font = input; initialization; preserveAspect; startGroup]
*)
RowSpaceBasis=Array[RowReduce[A][[#]]&,3]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
TableForm[{{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}]
;[o]
1   0   0

0   1   0

0   0   1
:[font = input; initialization; preserveAspect; startGroup]
*)
ColumnSpaceBasis=RowReduce[Transpose[A]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
TableForm[{{1, 0, 0, 1/20, 3/35, 3/28, 5/42, 1/8, 7/55, 
 
    7/55, 18/143, 45/364, 11/91, 33/280}, 
 
   {0, 1, 0, -3/5, -32/35, -15/14, -8/7, -7/6, -64/55, 
 
    -63/55, -160/143, -99/91, -96/91, -143/140}, 
 
   {0, 0, 1, 3/2, 12/7, 25/14, 25/14, 7/4, 56/33, 18/11, 
 
    225/143, 275/182, 132/91, 39/28}}]
;[o]
            1      3       3       5      1      7
            --     --      --      --     -      --
1   0   0   20     35      28      42     8      55
 
      7       18       45      11      33
      --      ---      ---     --      ---
      55      143      364     91      280

              3      32      15      8      7      64
            -(-)   -(--)   -(--)   -(-)   -(-)   -(--)
0   1   0     5      35      14      7      6      55
 
        63      160      99      96      143
      -(--)   -(---)   -(--)   -(--)   -(---)
        55      143      91      91      140

            3      12      25      25     7      56
            -      --      --      --     -      --
0   0   1   2      7       14      14     4      33
 
      18      225      275     132     39
      --      ---      ---     ---     --
      11      143      182     91      28
:[font = input; initialization; preserveAspect]
*)
c)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
A=Array[h,{27,5}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
TableForm[{{1, 1/2, 1/3, 1/4, 1/5}, 
 
   {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}, 
 
   {1/9, 1/10, 1/11, 1/12, 1/13}, 
 
   {1/10, 1/11, 1/12, 1/13, 1/14}, 
 
   {1/11, 1/12, 1/13, 1/14, 1/15}, 
 
   {1/12, 1/13, 1/14, 1/15, 1/16}, 
 
   {1/13, 1/14, 1/15, 1/16, 1/17}, 
 
   {1/14, 1/15, 1/16, 1/17, 1/18}, 
 
   {1/15, 1/16, 1/17, 1/18, 1/19}, 
 
   {1/16, 1/17, 1/18, 1/19, 1/20}, 
 
   {1/17, 1/18, 1/19, 1/20, 1/21}, 
 
   {1/18, 1/19, 1/20, 1/21, 1/22}, 
 
   {1/19, 1/20, 1/21, 1/22, 1/23}, 
 
   {1/20, 1/21, 1/22, 1/23, 1/24}, 
 
   {1/21, 1/22, 1/23, 1/24, 1/25}, 
 
   {1/22, 1/23, 1/24, 1/25, 1/26}, 
 
   {1/23, 1/24, 1/25, 1/26, 1/27}, 
 
   {1/24, 1/25, 1/26, 1/27, 1/28}, 
 
   {1/25, 1/26, 1/27, 1/28, 1/29}, 
 
   {1/26, 1/27, 1/28, 1/29, 1/30}, 
 
   {1/27, 1/28, 1/29, 1/30, 1/31}}]
;[o]
     1    1    1    1
     -    -    -    -
1    2    3    4    5

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

1    1    1    1    1
-    --   --   --   --
9    10   11   12   13

1    1    1    1    1
--   --   --   --   --
10   11   12   13   14

1    1    1    1    1
--   --   --   --   --
11   12   13   14   15

1    1    1    1    1
--   --   --   --   --
12   13   14   15   16

1    1    1    1    1
--   --   --   --   --
13   14   15   16   17

1    1    1    1    1
--   --   --   --   --
14   15   16   17   18

1    1    1    1    1
--   --   --   --   --
15   16   17   18   19

1    1    1    1    1
--   --   --   --   --
16   17   18   19   20

1    1    1    1    1
--   --   --   --   --
17   18   19   20   21

1    1    1    1    1
--   --   --   --   --
18   19   20   21   22

1    1    1    1    1
--   --   --   --   --
19   20   21   22   23

1    1    1    1    1
--   --   --   --   --
20   21   22   23   24

1    1    1    1    1
--   --   --   --   --
21   22   23   24   25

1    1    1    1    1
--   --   --   --   --
22   23   24   25   26

1    1    1    1    1
--   --   --   --   --
23   24   25   26   27

1    1    1    1    1
--   --   --   --   --
24   25   26   27   28

1    1    1    1    1
--   --   --   --   --
25   26   27   28   29

1    1    1    1    1
--   --   --   --   --
26   27   28   29   30

1    1    1    1    1
--   --   --   --   --
27   28   29   30   31
:[font = input; initialization; preserveAspect; startGroup]
*)
NullSpace[A]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{}
;[o]
{}
:[font = input; initialization; preserveAspect; startGroup]
*)
RowSpaceBasis=Array[RowReduce[A][[#]]&,5]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
TableForm[{{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]
*)
ColumnSpaceBasis=RowReduce[Transpose[A]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
TableForm[{{1, 0, 0, 0, 0, 1/252, 5/462, 5/264, 35/1287, 
 
    5/143, 6/143, 5/104, 165/3094, 55/952, 715/11628, 
 
    1001/15504, 65/969, 130/1881, 340/4807, 255/3542, 
 
    646/8855, 969/13156, 133/1794, 209/2808, 253/3393, 
 
    253/3393, 12650/169911}, 
 
   {0, 1, 0, 0, 0, -5/42, -24/77, -35/66, -320/429, 
 
    -135/143, -160/143, -33/26, -2160/1547, -715/476, 
 
    -1540/969, -2145/1292, -1664/969, -1105/627, -8640/4807, 
 
    -3230/1771, -3264/1771, -61047/32890, -1672/897, 
 
    -437/234, -704/377, -6325/3393, -105248/56637}, 
 
   {0, 0, 1, 0, 0, 5/6, 45/22, 147/44, 1960/429, 810/143, 
 
    945/143, 385/52, 1782/221, 585/68, 35035/3876, 
 
    24255/2584, 3120/323, 6188/627, 48195/4807, 2565/253, 
 
    2584/253, 67473/6578, 30723/2990, 4807/468, 3864/377, 
 
    3850/377, 82225/8091}, 
 
   {0, 0, 0, 1, 0, -20/9, -160/33, -245/33, -12544/1287, 
 
    -1680/143, -1920/143, -385/26, -3520/221, -286/17, 
 
    -50960/2907, -35035/1938, -17920/969, -35360/1881, 
 
    -91392/4807, -4845/253, -4864/253, -63308/3289, 
 
    -17248/897, -33649/1755, -64768/3393, -64400/3393, 
 
    -457600/24273}, {0, 0, 0, 0, 1, 5/2, 45/11, 245/44, 
 
    980/143, 1134/143, 1260/143, 495/52, 4455/442, 715/68, 
 
    7007/646, 28665/2584, 3640/323, 2380/209, 55080/4807, 
 
    2907/253, 2907/253, 75411/6578, 10241/897, 1771/156, 
 
    21252/1885, 12650/1131, 29900/2697}}]
;[o]
                     1       5        5        35
                    ---     ---      ---      ----
1   0   0   0   0   252     462      264      1287
 
       5         6         5       165       55
      ---       ---       ---      ----      ---
      143       143       104      3094      952
 
    715       1001       65         130        340
   -----      -----      ---        ----       ----
   11628      15504      969        1881       4807
 
      255       646        969       133        209
      ----      ----      -----      ----       ----
      3542      8855      13156      1794       2808
 
      253        253        12650
      ----       ----       ------
      3393       3393       169911

                      5       24       35       320
                    -(--)   -(--)    -(--)    -(---)
0   1   0   0   0     42      77       66       429
 
        135       160       33       2160      715
      -(---)    -(---)    -(--)    -(----)   -(---)
        143       143       26       1547      476
 
     1540       2145       1664       1105       8640
   -(----)    -(----)    -(----)    -(----)    -(----)
     969        1292       969        627        4807
 
        3230      3264      61047      1672       437
      -(----)   -(----)   -(-----)   -(----)    -(---)
        1771      1771      32890      897        234
 
        704        6325       105248
      -(---)     -(----)    -(------)
        377        3393       56637

                    5       45       147      1960
                    -       --       ---      ----
0   0   1   0   0   6       22       44       429
 
      810       945       385      1782      585
      ---       ---       ---      ----      ---
      143       143       52       221       68
 
   35035      24255      3120       6188       48195
   -----      -----      ----       ----       -----
   3876       2584       323        627        4807
 
      2565      2584      67473      30723      4807
      ----      ----      -----      -----      ----
      253       253       6578       2990       468
 
      3864       3850       82225
      ----       ----       -----
      377        377        8091

                      20      160      245      12544
                    -(--)   -(---)   -(---)   -(-----)
0   0   0   1   0     9       33       33       1287
 
        1680      1920      385      3520      286
      -(----)   -(----)   -(---)   -(----)   -(---)
        143       143       26       221       17
 
     50960      35035      17920      35360      91392
   -(-----)   -(-----)   -(-----)   -(-----)   -(-----)
     2907       1938        969       1881       4807
 
        4845      4864      63308      17248      33649
      -(----)   -(----)   -(-----)   -(-----)   -(-----)
        253       253       3289        897       1755
 
        64768      64400      457600
      -(-----)   -(-----)   -(------)
        3393       3393       24273

                    5       45       245      980
                    -       --       ---      ---
0   0   0   0   1   2       11       44       143
 
      1134      1260      495      4455      715
      ----      ----      ---      ----      ---
      143       143       52       442       68
 
   7007       28665      3640       2380       55080
   ----       -----      ----       ----       -----
   646        2584       323        209        4807
 
      2907      2907      75411      10241      1771
      ----      ----      -----      -----      ----
      253       253       6578        897       156
 
      21252      12650      29900
      -----      -----      -----
      1885       1131       2697
^*)