4001-Alg-Alg-Alg

︠4ca5609b-bfb9-45b0-8a0c-821d73fc7038i︠
%html
<h3>Alg&egrave;bre lin&eacute;aire</h3>
<p>Sage propose de nombreuses possibilit&eacute; pour l'alg&egrave;bre lin&eacute;aire</p>

︡cde60a60-328f-483e-a1e5-f01dd466403d︡{"html": "<h3>Alg&egrave;bre lin&eacute;aire</h3>\n<p>Sage propose de nombreuses possibilit&eacute; pour l'alg&egrave;bre lin&eacute;aire</p>"}︡
︠6d87d7a7-3ac0-487c-9335-3a443e85d8cd︠
A = Matrix(QQ, [[0, -1, -1, 1], [1, 1, 1, 1], [2, 4, 1, -2], [3, 1, -2, 2]])
print("Matrix A:")
show(A)

# Getting elements of a matrix
print("A[0] = {0}".format(A[0]))
print("A[1, 2] = {0}".format(A[1, 2]))
print("A[2, 1] = {0}".format(A[2, 1]))
print("A[0:2]")
show(A[0:2])
print("A[0, 2:4] = {0}".format(A[0, 2:4]))
print("A[:,0]:")
show(A[:,0])

# Getting parts of a matrix
print("Third row:")
print(A.row(2))
print("Second column:")
print(A.column(1))
print("Lower right submatrix:")
show(A.submatrix(2, 2, 2, 2))
︡a708391c-a45e-406b-90c7-ed721bf19bf2︡{"stdout": "Matrix A:"}︡{"html": "<div class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n0 & -1 & -1 & 1 \\\\\n1 & 1 & 1 & 1 \\\\\n2 & 4 & 1 & -2 \\\\\n3 & 1 & -2 & 2\n\\end{array}\\right)</div>"}︡{"stdout": "A[0] = (0, -1, -1, 1)\nA[1, 2] = 1\nA[2, 1] = 4\nA[0:2]"}︡{"html": "<div class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n0 & -1 & -1 & 1 \\\\\n1 & 1 & 1 & 1\n\\end{array}\\right)</div>"}︡{"stdout": "A[0, 2:4] = [-1  1]\nA[:,0]:"}︡{"html": "<div class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r}\n0 \\\\\n1 \\\\\n2 \\\\\n3\n\\end{array}\\right)</div>"}︡{"stdout": "Third row:\n(2, 4, 1, -2)\nSecond column:\n(-1, 1, 4, 1)\nLower right submatrix:"}︡{"html": "<div class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rr}\n1 & -2 \\\\\n-2 & 2\n\\end{array}\\right)</div>"}︡
︠bc361529-f261-4f59-b0ce-5733ff862b46︠
A = Matrix(QQ, [[1, 2, 3], [4, 5, 6], [7, 8, 9]])
print("Matrix A:")
show(A)

# Elementary row operations
print("Scaling second row by two:")
A.rescale_row(1, 2)
show(A)

print("Swapping first and second rows:")
A.swap_rows(0, 1)
show(A)

print("Adding 3*(row 1) to row 0:")
A.add_multiple_of_row(0, 1 ,3)
show(A)

print("A in echelon form:")
show(A.echelon_form())
︡1b93fcb1-c9f2-4096-9d76-91d0c96a71aa︡{"stdout": "Matrix A:"}︡{"html": "<div class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrr}\n1 & 2 & 3 \\\\\n4 & 5 & 6 \\\\\n7 & 8 & 9\n\\end{array}\\right)</div>"}︡{"stdout": "Scaling second row by two:"}︡{"html": "<div class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrr}\n1 & 2 & 3 \\\\\n8 & 10 & 12 \\\\\n7 & 8 & 9\n\\end{array}\\right)</div>"}︡{"stdout": "Swapping first and second rows:"}︡{"html": "<div class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrr}\n8 & 10 & 12 \\\\\n1 & 2 & 3 \\\\\n7 & 8 & 9\n\\end{array}\\right)</div>"}︡{"stdout": "Adding 3*(row 1) to row 0:"}︡{"html": "<div class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrr}\n11 & 16 & 21 \\\\\n1 & 2 & 3 \\\\\n7 & 8 & 9\n\\end{array}\\right)</div>"}︡{"stdout": "A in echelon form:"}︡{"html": "<div class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrr}\n1 & 0 & -1 \\\\\n0 & 1 & 2 \\\\\n0 & 0 & 0\n\\end{array}\\right)</div>"}︡
︠fed8af4a-10ef-467b-9227-571b7291e29c︠
M3 = MatrixSpace(QQ, 3, 2)
A = M3.matrix([[3, 2, 1], [4, 5, 6]])
B = M3.matrix([[2, 2, 2], [1, 2, 3]])
print("Matrix addition:")
show(A + B)

print("Scalar multiplication:")
show(1/2 * A)

var('a b c d e f')
C = Matrix(QQ, [[4, 2, 1], [5, 3, 7]])
D = Matrix(SR, [[a, b], [c, d], [e, f]])
print("Matrix multiplication:")
show(C * D)

var('x1 x2 x3')
X = vector([x1,x2,x3])
print("Multiplying a matrix and a vector:")
show(C * X)
︡63401b32-0f3c-4582-b507-065387051bb0︡{"stdout": "Matrix addition:"}︡{"html": "<div class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rr}\n5 & 4 \\\\\n3 & 5 \\\\\n7 & 9\n\\end{array}\\right)</div>"}︡{"stdout": "Scalar multiplication:"}︡{"html": "<div class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rr}\n\\frac{3}{2} & 1 \\\\\n\\frac{1}{2} & 2 \\\\\n\\frac{5}{2} & 3\n\\end{array}\\right)</div>"}︡{"stdout": "Matrix multiplication:"}︡{"html": "<div class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rr}\n4 \\, a + 2 \\, c + e & 4 \\, b + 2 \\, d + f \\\\\n5 \\, a + 3 \\, c + 7 \\, e & 5 \\, b + 3 \\, d + 7 \\, f\n\\end{array}\\right)</div>"}︡{"stdout": "Multiplying a matrix and a vector:"}︡{"html": "<div class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(4 \\, x_{1} + 2 \\, x_{2} + x_{3},\\,5 \\, x_{1} + 3 \\, x_{2} + 7 \\, x_{3}\\right)</div>"}︡
︠2a44f8ef-3281-4217-8fa2-ab149af90d58︠
A = matrix(QQ, [[2, 5, 4], [3, 1, 2], [5, 4, 6]])
print("Matrix A:")
show(A)

# Scalar operations
print("Determinant of A: {0}".format(A.det()))
print("Rank of A: {0}".format(A.rank()))
print("Euclidean norm: {0}".format(A.norm()))
print("Frobenius norm: {0}".format(A.norm('frob')))

# Matrix operations
print("Transpose of A:")
show(A.transpose())

print("Inverse of A:")
show(A.inverse())

print("Adjoint of A:")
show(A.adjoint())

print("Testing adj(A)/det(A) == inverse(A)")
A.adjoint()/A.det() == A.inverse()
︡ec2b3020-9470-407f-8a34-cb8361340772︡{"stdout": "Matrix A:"}︡{"html": "<div class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrr}\n2 & 5 & 4 \\\\\n3 & 1 & 2 \\\\\n5 & 4 & 6\n\\end{array}\\right)</div>"}︡{"stdout": "Determinant of A: -16\nRank of A: 3\nEuclidean norm: 11.3469601386\nFrobenius norm: 11.6619037897\nTranspose of A:"}︡{"html": "<div class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrr}\n2 & 3 & 5 \\\\\n5 & 1 & 4 \\\\\n4 & 2 & 6\n\\end{array}\\right)</div>"}︡{"stdout": "Inverse of A:"}︡{"html": "<div class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrr}\n\\frac{1}{8} & \\frac{7}{8} & -\\frac{3}{8} \\\\\n\\frac{1}{2} & \\frac{1}{2} & -\\frac{1}{2} \\\\\n-\\frac{7}{16} & -\\frac{17}{16} & \\frac{13}{16}\n\\end{array}\\right)</div>"}︡{"stdout": "Adjoint of A:"}︡{"html": "<div class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrr}\n-2 & -14 & 6 \\\\\n-8 & -8 & 8 \\\\\n7 & 17 & -13\n\\end{array}\\right)</div>"}︡{"stdout": "Testing adj(A)/det(A) == inverse(A)\nTrue"}︡