Math 1062 -- Feb 13, 2012 -- Linear Algebra (part 1)

Questions? Reminder: the "short" take-home midterm will be assigned Wed and due Friday at midnight.
Warm Up: http://aleph.sagemath.org, persistent URL's, and embedding Sage (including interacts) in your own web page. See Beezer's examples, the official directions (22 easy steps?), and this simple complete example.
Warm Up for today: Fast 2x2 matrix multiplication? Talk for a moment about this sage-support thread.
General overview of linear algebra in Sage.

Making a Matrix

[1,2; 2,3; pi,e^2]

A = matrix([[1,2], [2,3], [pi, e^2]]); A.base_ring()

Symbolic Ring

Symbolic Ring

K = GF(7); K

Finite Field of size 7

Finite Field of size 7

list(K)

[0, 1, 2, 3, 4, 5, 6]

[0, 1, 2, 3, 4, 5, 6]

RDF

Real Double Field

Real Double Field

S.<chris2, william2> = RR[]

random_matrix(S, 2)

[         -0.821876689273126*chris2^2 -
0.337570915811116*chris2*william2 - 0.445009814043026*william2^2 -
0.933733757961343*chris2 - 0.328938818266539 -0.901307772747002*chris2^2
- 0.343913151855248*chris2*william2 - 0.955029600171170*william2^2 -
0.613473476443475*chris2 + 0.817179944585997*william2]
[0.692008619961593*chris2^2 - 0.666991104945709*chris2*william2 -
0.0950703419495393*william2^2 + 0.261926742846123*chris2 -
0.801579599176361*william2                  0.143659682615950*chris2^2 -
0.221007018170083*william2^2 - 0.162863097602838*chris2 -
0.617597598526627*william2 - 0.553651224948270]

[         -0.821876689273126*chris2^2 - 0.337570915811116*chris2*william2 - 0.445009814043026*william2^2 - 0.933733757961343*chris2 - 0.328938818266539 -0.901307772747002*chris2^2 - 0.343913151855248*chris2*william2 - 0.955029600171170*william2^2 - 0.613473476443475*chris2 + 0.817179944585997*william2]
[0.692008619961593*chris2^2 - 0.666991104945709*chris2*william2 - 0.0950703419495393*william2^2 + 0.261926742846123*chris2 - 0.801579599176361*william2                  0.143659682615950*chris2^2 - 0.221007018170083*william2^2 - 0.162863097602838*chris2 - 0.617597598526627*william2 - 0.553651224948270]

(chris2 + william2)^3

chris2^3 + 3.00000000000000*chris2^2*william2 +
3.00000000000000*chris2*william2^2 + william2^3

chris2^3 + 3.00000000000000*chris2^2*william2 + 3.00000000000000*chris2*william2^2 + william2^3

S = RR['chris, william']; S

Multivariate Polynomial Ring in chris, william over Real Field with 53
bits of precision

Multivariate Polynomial Ring in chris, william over Real Field with 53 bits of precision

S.0

chris

chris

S.1

william

william

S.gens()

(chris, william)

(chris, william)

S.gen(0)

chris

chris

chris

Traceback (click to the left of this block for traceback)
...
NameError: name 'chris' is not defined

Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "_sage_input_92.py", line 10, in <module>
    exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("Y2hyaXM="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))
  File "", line 1, in <module>
    
  File "/private/var/folders/4z/h5m3q9b94zs641jlbxj79sv00000gn/T/tmp1kumAa/___code___.py", line 2, in <module>
    exec compile(u'chris
  File "", line 1, in <module>
    
NameError: name 'chris' is not defined

A = matrix(GF(7), 3, 2, [1,2,3]*2); A

[1 2]
[3 1]
[2 3]

[1 2]
[3 1]
[2 3]

A.base_ring()

Finite Field of size 7

Finite Field of size 7

A = matrix(8, [1..8^2]) A

[ 1  2  3  4  5  6  7  8]
[ 9 10 11 12 13 14 15 16]
[17 18 19 20 21 22 23 24]
[25 26 27 28 29 30 31 32]
[33 34 35 36 37 38 39 40]
[41 42 43 44 45 46 47 48]
[49 50 51 52 53 54 55 56]
[57 58 59 60 61 62 63 64]

[ 1  2  3  4  5  6  7  8]
[ 9 10 11 12 13 14 15 16]
[17 18 19 20 21 22 23 24]
[25 26 27 28 29 30 31 32]
[33 34 35 36 37 38 39 40]
[41 42 43 44 45 46 47 48]
[49 50 51 52 53 54 55 56]
[57 58 59 60 61 62 63 64]

A.base_ring()

Integer Ring

Integer Ring

v = A[1]; type(v)

<type 'sage.modules.vector_integer_dense.Vector_integer_dense'>

<type 'sage.modules.vector_integer_dense.Vector_integer_dense'>

A.column(1)

(2, 2, 1)

(2, 2, 1)

A.row(1)

(4/5, 2, 7)

(4/5, 2, 7)

v.dot_product(v)

1341/25

1341/25

3.5 * v

(2.80000000000000, 7.00000000000000, 24.5000000000000)

(2.80000000000000, 7.00000000000000, 24.5000000000000)

[  1   2   3]
[4/5   2   7]
[  1   1   1]

[  1   2   3]
[4/5   2   7]
[  1   1   1]

[ 1  2  3  4  5  6  7  8]
[ 9 10 11 12 13 14 15 16]
[17 18 19 20 21 22 23 24]
[25 26 27 28 29 30 31 32]
[33 34 35 36 37 38 39 40]
[41 42 43 44 45 46 47 48]
[49 50 51 52 53 54 55 56]
[57 58 59 60 61 62 63 64]

[ 1  2  3  4  5  6  7  8]
[ 9 10 11 12 13 14 15 16]
[17 18 19 20 21 22 23 24]
[25 26 27 28 29 30 31 32]
[33 34 35 36 37 38 39 40]
[41 42 43 44 45 46 47 48]
[49 50 51 52 53 54 55 56]
[57 58 59 60 61 62 63 64]

A[:,1::2]

[ 2  4  6  8]
[10 12 14 16]
[18 20 22 24]
[26 28 30 32]
[34 36 38 40]
[42 44 46 48]
[50 52 54 56]
[58 60 62 64]

[ 2  4  6  8]
[10 12 14 16]
[18 20 22 24]
[26 28 30 32]
[34 36 38 40]
[42 44 46 48]
[50 52 54 56]
[58 60 62 64]

A[0,1]

A[0,1] = 2012

A.set_immutable()

0,1,2

(0, 1, 2)

(0, 1, 2)

A[0,1] = 92090834

Traceback (click to the left of this block for traceback)
...
ValueError: matrix is immutable; please change a copy instead (i.e., use
copy(M) to change a copy of M).

Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "_sage_input_142.py", line 10, in <module>
    exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("QVswLDFdID0gOTIwOTA4MzQ="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))
  File "", line 1, in <module>
    
  File "/private/var/folders/4z/h5m3q9b94zs641jlbxj79sv00000gn/T/tmpHwLvwE/___code___.py", line 3, in <module>
    exec compile(u'A[_sage_const_0 ,_sage_const_1 ] = _sage_const_92090834 
  File "", line 1, in <module>
    
  File "matrix0.pyx", line 1339, in sage.matrix.matrix0.Matrix.__setitem__ (sage/matrix/matrix0.c:5846)
  File "matrix0.pyx", line 404, in sage.matrix.matrix0.Matrix.check_mutability (sage/matrix/matrix0.c:3884)
ValueError: matrix is immutable; please change a copy instead (i.e., use copy(M) to change a copy of M).

v = A[0]; v

(1, 2012, 3, 4, 5, 6, 7, 8)

(1, 2012, 3, 4, 5, 6, 7, 8)

v[1] = 193939

Traceback (click to the left of this block for traceback)
...
ValueError: vector is immutable; please change a copy instead (use
copy())

Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "_sage_input_139.py", line 10, in <module>
    exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("dlsxXSA9IDE5MzkzOQ=="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))
  File "", line 1, in <module>
    
  File "/private/var/folders/4z/h5m3q9b94zs641jlbxj79sv00000gn/T/tmpv4ZPc_/___code___.py", line 3, in <module>
    exec compile(u'v[_sage_const_1 ] = _sage_const_193939 
  File "", line 1, in <module>
    
  File "vector_integer_dense.pyx", line 157, in sage.modules.vector_integer_dense.Vector_integer_dense.__setitem__ (sage/modules/vector_integer_dense.c:3008)
ValueError: vector is immutable; please change a copy instead (use copy())

(A[0])[1] = 193939

Traceback (click to the left of this block for traceback)
...
ValueError: vector is immutable; please change a copy instead (use
copy())

Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "_sage_input_136.py", line 10, in <module>
    exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("QVswXVsxXSA9IDE5MzkzOQ=="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))
  File "", line 1, in <module>
    
  File "/private/var/folders/4z/h5m3q9b94zs641jlbxj79sv00000gn/T/tmp3B0USk/___code___.py", line 3, in <module>
    exec compile(u'A[_sage_const_0 ][_sage_const_1 ] = _sage_const_193939 
  File "", line 1, in <module>
    
  File "vector_integer_dense.pyx", line 157, in sage.modules.vector_integer_dense.Vector_integer_dense.__setitem__ (sage/modules/vector_integer_dense.c:3008)
ValueError: vector is immutable; please change a copy instead (use copy())

[   1 2012    3    4    5    6    7    8]
[   9   10   11   12   13   14   15   16]
[  17   18   19   20   21   22   23   24]
[  25   26   27   28   29   30   31   32]
[  33   34   35   36   37   38   39   40]
[  41   42   43   44   45   46   47   48]
[  49   50   51   52   53   54   55   56]
[  57   58   59   60   61   62   63   64]

[   1 2012    3    4    5    6    7    8]
[   9   10   11   12   13   14   15   16]
[  17   18   19   20   21   22   23   24]
[  25   26   27   28   29   30   31   32]
[  33   34   35   36   37   38   39   40]
[  41   42   43   44   45   46   47   48]
[  49   50   51   52   53   54   55   56]
[  57   58   59   60   61   62   63   64]

A*A

[10061/2515  6039/2012 12079/2012]
[      47/5 27666/2515      117/5]
[      14/5  6037/2012         11]

[10061/2515  6039/2012 12079/2012]
[      47/5 27666/2515      117/5]
[      14/5  6037/2012         11]

A.echelon_form()

[1 0 0]
[0 1 0]
[0 0 1]

[1 0 0]
[0 1 0]
[0 0 1]

show(A)

\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrr}
1 & \frac{1}{2012} & 3 \\
\frac{4}{5} & 2 & 7 \\
1 & 1 & 1
\end{array}\right)

\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrr}
1 & \frac{1}{2012} & 3 \\
\frac{4}{5} & 2 & 7 \\
1 & 1 & 1
\end{array}\right)

latex(A)

\left(\begin{array}{rrr}
1 & \frac{1}{2012} & 3 \\
\frac{4}{5} & 2 & 7 \\
1 & 1 & 1
\end{array}\right)

\left(\begin{array}{rrr}
1 & \frac{1}{2012} & 3 \\
\frac{4}{5} & 2 & 7 \\
1 & 1 & 1
\end{array}\right)

A.charpoly()

x^3 - 4*x^2 - 12576/2515*x + 17297/2012

x^3 - 4*x^2 - 12576/2515*x + 17297/2012

A.det()

-17297/2012

-17297/2012

A.solve_right(vector([1,2,3]))

(34185/17297, 116696/86485, -28166/86485)

(34185/17297, 116696/86485, -28166/86485)

A.solve_left(vector([1,2,3]))

(-38228/86485, 8046/17297, 92529/86485)

(-38228/86485, 8046/17297, 92529/86485)

show(A.transpose())

\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrr}
1 & \frac{4}{5} & 1 \\
\frac{1}{2012} & 2 & 1 \\
3 & 7 & 1
\end{array}\right)

\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrr}
1 & \frac{4}{5} & 1 \\
\frac{1}{2012} & 2 & 1 \\
3 & 7 & 1
\end{array}\right)

Very powerful!

A = random_matrix(ZZ, 500) time v = A.solve_right(vector([1..500]))

Time: CPU 2.68 s, Wall: 1.86 s

Time: CPU 2.68 s, Wall: 1.86 s

len(str(v))

Explain some of the examples in the docstring below:

matrix?

Math 1062 -- Feb 13, 2012 -- Linear Algebra (part 1)

92 days ago by William_Stein

Math 1062 -- Feb 13, 2012 -- Linear Algebra (part 1)

Making a Matrix