This document is one of More SageMath Tutorials. You may edit it on github. \(\def\NN{\mathbb{N}}\) \(\def\ZZ{\mathbb{Z}}\) \(\def\QQ{\mathbb{Q}}\) \(\def\RR{\mathbb{R}}\) \(\def\CC{\mathbb{C}}\)

Linear Algebra¶

Vectors¶

To create a vector in Sage, use the vector command.

Note

vectors in Sage are row vectors!

Exercises

Create the vector \(x = (1, 2, \ldots, 100)\).
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Create the vector \(y = (1^2, 2^2, \ldots, 100^2)\).
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Type x. and hit tab to see the available methods for vectors. Find the norm (length) of the vectors x and y.
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Find the dot product of x and y .
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[The above exercises are essentially the first problem on Exercise Set 1 of William Stein’s Math 480b]

Matrices¶

Use the matrix command to create the following matrix over the rational numbers (hint: in Sage, QQ denotes the field of rational numbers).

\[\begin{split}\left(\begin{array}{rrrrrr}3 & 2 & 2 & 1 & 1 & 0 \\2 & 3 & 1 & 0 & 2 & 1 \\2 & 1 & 3 & 2 & 0 & 1 \\1 & 0 & 2 & 3 & 1 & 2 \\1 & 2 & 0 & 1 & 3 & 2 \\0 & 1 & 1 & 2 & 2 & 3\end{array}\right)\end{split}\]
1. Find the echelon form of the above matrix.
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2. Find the right kernel of the matrix.
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3. Find the eigenvalues of the matrix.
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4. Find the left eigenvectors of the matrix.
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5. Find the right eigenspaces of the matrix.
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For what values of \(k\) is the determinant of the following matrix \(0\)?

\[\begin{split}\left(\begin{array}{rrr}1 & 1 & -1 \\2 & 3 & k \\1 & k & 3\end{array}\right)\end{split}\]
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[K. R. Matthews, Elementary Linear Algebra , Chapter 4, Problem 8]
Prove that the determinant of the following matrix is \(-8\).

\[\begin{split}\left(\begin{array}{rrr}{n}^{2} & {\left( n + 1 \right)}^{2} & {\left( n + 2\right)}^{2} \\{\left( n + 1 \right)}^{2} & {\left( n + 2 \right)}^{2} &{\left( n + 3 \right)}^{2} \\{\left( n + 2 \right)}^{2} & {\left( n + 3 \right)}^{2} & {\left( n + 4 \right)}^{2}\end{array}\right)\end{split}\]
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[K. R. Matthews, Elementary Linear Algebra , Chapter 4, Problem 3]
Prove that if \(a \neq c\), then the line through the points \((a,b)\) and \((c,d)\) is given by the following equation.

\[\begin{split}\det\left(\begin{array}{rrr}x & y & 1 \\a & b & 1 \\c & d & 1\end{array}\right) = 0.\end{split}\]
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Find the determinant of the following matrices.

\[\begin{split}\left(\begin{array}{r}1\end{array}\right),\left(\begin{array}{rr}1 & 1 \\r & 1\end{array}\right),\left(\begin{array}{rrr}1 & 1 & 1 \\r & 1 & 1 \\r & r & 1\end{array}\right),\left(\begin{array}{rrrr}1 & 1 & 1 & 1 \\r & 1 & 1 & 1 \\r & r & 1 & 1 \\r & r & r & 1\end{array}\right),\left(\begin{array}{rrrrr}1 & 1 & 1 & 1 & 1 \\r & 1 & 1 & 1 & 1 \\r & r & 1 & 1 & 1 \\r & r & r & 1 & 1 \\r & r & r & r & 1\end{array}\right)\end{split}\]

Make a conjecture about the determinant of an arbitrary matrix in this sequence. Can you prove it your conjecture?
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[Adapted from: K. R. Matthews, Elementary Linear Algebra , Chapter 4, Problem 19]

What is the largest determinant possible for a \(3\times3\) matrix whose entries are \(1, 2, \dots, 9\) (each occurring exactly once, in any order). How many matrices \(M\) achieve this maximum?

(Hint: You might find the command Permutations useful. The following code will construct all the lists that have the entries \(1, 2, 3, 4\), each appearing exactly once.)

for P in Permutations(4):
    L = list(P)
    print L

sage: for P in Permutations(4):
....:     L = list(P)
....:     print L
[1, 2, 3, 4]
[1, 2, 4, 3]
[1, 3, 2, 4]
[1, 3, 4, 2]
[1, 4, 2, 3]
[1, 4, 3, 2]
[2, 1, 3, 4]
[2, 1, 4, 3]
[2, 3, 1, 4]
[2, 3, 4, 1]
[2, 4, 1, 3]
[2, 4, 3, 1]
[3, 1, 2, 4]
[3, 1, 4, 2]
[3, 2, 1, 4]
[3, 2, 4, 1]
[3, 4, 1, 2]
[3, 4, 2, 1]
[4, 1, 2, 3]
[4, 1, 3, 2]
[4, 2, 1, 3]
[4, 2, 3, 1]
[4, 3, 1, 2]
[4, 3, 2, 1]

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Project Euler Problem 11 ¶

In the \(20 \times 20\) grid below, four numbers along a diagonal line have been highlighted.

08 02 22 97 38 15 00 40 00 75 04 05 07 78 52 12 50 77 91 08

49 49 99 40 17 81 18 57 60 87 17 40 98 43 69 48 04 56 62 00

81 49 31 73 55 79 14 29 93 71 40 67 53 88 30 03 49 13 36 65

52 70 95 23 04 60 11 42 69 24 68 56 01 32 56 71 37 02 36 91

22 31 16 71 51 67 63 89 41 92 36 54 22 40 40 28 66 33 13 80

24 47 32 60 99 03 45 02 44 75 33 53 78 36 84 20 35 17 12 50

32 98 81 28 64 23 67 10 26 38 40 67 59 54 70 66 18 38 64 70

67 26 20 68 02 62 12 20 95 63 94 39 63 08 40 91 66 49 94 21

24 55 58 05 66 73 99 26 97 17 78 78 96 83 14 88 34 89 63 72

21 36 23 09 75 00 76 44 20 45 35 14 00 61 33 97 34 31 33 95

78 17 53 28 22 75 31 67 15 94 03 80 04 62 16 14 09 53 56 92

16 39 05 42 96 35 31 47 55 58 88 24 00 17 54 24 36 29 85 57

86 56 00 48 35 71 89 07 05 44 44 37 44 60 21 58 51 54 17 58

19 80 81 68 05 94 47 69 28 73 92 13 86 52 17 77 04 89 55 40

04 52 08 83 97 35 99 16 07 97 57 32 16 26 26 79 33 27 98 66

88 36 68 87 57 62 20 72 03 46 33 67 46 55 12 32 63 93 53 69

04 42 16 73 38 25 39 11 24 94 72 18 08 46 29 32 40 62 76 36

20 69 36 41 72 30 23 88 34 62 99 69 82 67 59 85 74 04 36 16

20 73 35 29 78 31 90 01 74 31 49 71 48 86 81 16 23 57 05 54

01 70 54 71 83 51 54 69 16 92 33 48 61 43 52 01 89 19 67 48

The product of these numbers is \(26 \times 63 \times 78 \times 14 = 1788696\).

What is the greatest product of four adjacent numbers in any direction (up, down, left, right, or diagonally) in the \(20 \times 20\) grid?

sage: A = matrix(20, 20, [
....:    8, 2,22,97,38,15, 0,40, 0,75, 4, 5, 7,78,52,12,50,77,91, 8,
....:    49,49,99,40,17,81,18,57,60,87,17,40,98,43,69,48, 4,56,62, 0,
....:    81,49,31,73,55,79,14,29,93,71,40,67,53,88,30, 3,49,13,36,65,
....:    52,70,95,23, 4,60,11,42,69,24,68,56, 1,32,56,71,37, 2,36,91,
....:    22,31,16,71,51,67,63,89,41,92,36,54,22,40,40,28,66,33,13,80,
....:    24,47,32,60,99, 3,45, 2,44,75,33,53,78,36,84,20,35,17,12,50,
....:    32,98,81,28,64,23,67,10,26,38,40,67,59,54,70,66,18,38,64,70,
....:    67,26,20,68, 2,62,12,20,95,63,94,39,63, 8,40,91,66,49,94,21,
....:    24,55,58, 5,66,73,99,26,97,17,78,78,96,83,14,88,34,89,63,72,
....:    21,36,23, 9,75, 0,76,44,20,45,35,14, 0,61,33,97,34,31,33,95,
....:    78,17,53,28,22,75,31,67,15,94, 3,80, 4,62,16,14, 9,53,56,92,
....:    16,39, 5,42,96,35,31,47,55,58,88,24, 0,17,54,24,36,29,85,57,
....:    86,56, 0,48,35,71,89, 7, 5,44,44,37,44,60,21,58,51,54,17,58,
....:    19,80,81,68, 5,94,47,69,28,73,92,13,86,52,17,77, 4,89,55,40,
....:    4,52, 8,83,97,35,99,16, 7,97,57,32,16,26,26,79,33,27,98,66,
....:    88,36,68,87,57,62,20,72, 3,46,33,67,46,55,12,32,63,93,53,69,
....:    4,42,16,73,38,25,39,11,24,94,72,18, 8,46,29,32,40,62,76,36,
....:    20,69,36,41,72,30,23,88,34,62,99,69,82,67,59,85,74, 4,36,16,
....:    20,73,35,29,78,31,90, 1,74,31,49,71,48,86,81,16,23,57, 5,54,
....:    1,70,54,71,83,51,54,69,16,92,33,48,61,43,52, 1,89,19,67,48
....:    ])

Linear Algebra¶

Vectors¶

Matrices¶

Project Euler Problem 11 ¶

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Linear Algebra¶

Vectors¶

Matrices¶

Project Euler Problem 11¶

Project Euler Problem 11 ¶