### Related

x: matrix ([1,2,2], ...

y: gramschmidt (x);

Calculate

x: matrix ([1, 2, 3]...

y: gramschmidt (x);

Calculate

x: matrix ([0, 1, 1,...

y: gramschmidt (x);

Calculate

v1: vector(1, 2);

v2: vector(1, -2);

Calculate

x: matrix ([-1,5,6],...

y: gramschmidt (x);

Calculate

x: matrix ([1, 2, -2...

y: gramschmidt (x);

Calculate

x: matrix ([-1, 1, 0...

y: gramschmidt (x);

Calculate

x: matrix ([1, 2, 3],...

y: gramschmidt (x);

Calculate

x: matrix ([1, 0, 0,...

y: gramschmidt (x);

Calculate

x: matrix ([1,2,1], ...

y: gramschmidt (x);

Calculate

### gramschmidt

Run Example
```(%i1)load (eigen);
(%o1)           /usr/share/maxima/5.21.1/share/matrix/eigen.mac
(%i2)  x: matrix ([2,1,1,1], [1,0,1,1], [-2,-1,0,-1]);
[  2    1   1   1  ]
[                  ]
(%o2)                        [  1    0   1   1  ]
[                  ]
[ - 2  - 1  0  - 1 ]
(%i3)  y: gramschmidt (x);
2
1    2   3  3      1  1  3    2
(%o3)         [[2, 1, 1, 1], [- -, - --, -, -], [- -, -, -, - -]]
7    7   7  7      5  5  5    5
(%i4)  map (innerproduct, [y[1], y[2], y[3]], [y[2], y[3], y[1]]);
(%o4)                              [0, 0, 0]
(%o5)           /usr/share/maxima/5.21.1/share/matrix/eigen.mac
(%i6)  ip (f, g) := integrate (f * g, u, a, b);
(%o6)                 ip(f, g) := integrate(f g, u, a, b)
(%i7)  y : gramschmidt ([1, sin(u), cos(u)], ip), a= -%pi/2, b=%pi/2;
%pi cos(u) - 2
(%o7)                     [1, sin(u), --------------]
%pi
(%i8)  map (ip, [y[1], y[2], y[3]], [y[2], y[3], y[1]]), a= -%pi/2, b=%pi/2;
(%o8)                              [0, 0, 0]
(%i9) ```
Run Example
```load (eigen);
(%o1)           /usr/share/maxima/5.21.1/share/matrix/eigen.mac
(%i2)  x: matrix ([1, 4, 8], [2,0,2], [0,5,1],[3,8,6]);
[ 1  4  8 ]
[         ]
[ 2  0  2 ]
(%o2)                             [         ]
[ 0  5  1 ]
[         ]
[ 3  8  6 ]
(%i3)  y: gramschmidt (x);
4     3         2              2
2     2   2     2  31  7 31    2  31
(%o3)   [[1, 4, 8], [--, - --, --], [-----, ----, - -----], [0, 0, 0]]
2     2   2      4      4       4
3     3   3      3      3       3
(%i4)  map (innerproduct, [y[1], y[2], y[3]], [y[2], y[3], y[1]]);
(%o4)                              [0, 0, 0]
(%o5)           /usr/share/maxima/5.21.1/share/matrix/eigen.mac
(%i6)  ip (f, g) := integrate (f * g, u, a, b);
(%o6)                 ip(f, g) := integrate(f g, u, a, b)
(%i7)  y : gramschmidt ([1, sin(u), cos(u)], ip), a= -%pi/2, b=%pi/2;
%pi cos(u) - 2
(%o7)                     [1, sin(u), --------------]
%pi
(%i8)  map (ip, [y[1], y[2], y[3]], [y[2], y[3], y[1]]), a= -%pi/2, b=%pi/2;
(%o8)                              [0, 0, 0]
(%i9) ```
Run Example
```load (eigen);
(%o1)           /usr/share/maxima/5.21.1/share/matrix/eigen.mac
(%i2)  x: matrix ([64,9,17,40,32,41,49,8], [2,55,47,26,34,23,15,58], [3,54,46,27,35,22,14,59]);
[ 64  9   17  40  32  41  49  8  ]
[                                ]
(%o2)                 [ 2   55  47  26  34  23  15  58 ]
[                                ]
[ 3   54  46  27  35  22  14  59 ]
(%i3)  y: gramschmidt (x);
(%o3) [[64, 9, 17, 40, 32, 41, 49, 8],
3          3                                3  3        3
2 42727  2  89 199  2  3 4493  2 7 1151  2 5 19 263  2  3  29    2  3449
[- -------, ---------, ---------, --------, ----------, --------, - -------,
2         2          2         2          2          2           2
53        53         53        53         53         53          53
2                2               2                2
2 17 4457    2  5 7 13 191    2  5 13 1523    2  3 5 13 19 29  2  5 11 13 157
---------], [-------------, - ------------, - ---------------, --------------,
2         31 61 257       31 61 257         31 61 257       31 61 257
53
2                2  2              2              2
2  3 5 13 619    2  3  5 13 227    2  5 13 41 53  2  3 5 7 13 107
-------------, - --------------, - -------------, ---------------]]
31 61 257        31 61 257         31 61 257       31 61 257
(%i4)  map (innerproduct, [y[1], y[2], y[3]], [y[2], y[3], y[1]]);
(%o4)                              [0, 0, 0]
(%o5)           /usr/share/maxima/5.21.1/share/matrix/eigen.mac
(%i6)  ip (f, g) := integrate (f * g, u, a, b);
(%o6)                 ip(f, g) := integrate(f g, u, a, b)
(%i7)  y : gramschmidt ([1, sin(u), cos(u)], ip), a= -%pi/2, b=%pi/2;
%pi cos(u) - 2
(%o7)                     [1, sin(u), --------------]
%pi
(%i8)  map (ip, [y[1], y[2], y[3]], [y[2], y[3], y[1]]), a= -%pi/2, b=%pi/2;
(%o8)                              [0, 0, 0]
(%i9) ```

### Related Help

Help for Gramschmidt