def matrix_producten():
# pylint: disable=C0103
RNG().set(4)
u = random_tensor(r"\vec u", 3, ret=True, details=False)
RNG().consume_entropy(0x02, -0x14, 0x14)
M = random_tensor(r"\mathbf{M}", (3,2), ret=True, details=False)
N = random_tensor(r"\mathbf{N}", (2,3), ret=True, details=False)
O = random_tensor(r"\mathbf{O}", (2,2), ret=True, details=False)
display(Markdown("<hr>"))
latex_bmatrix(O.dot(N.dot(u)), r"\mathbf{O} (\mathbf{N} \vec u)", details=False)
latex_bmatrix(O.dot(N).dot(u), r"(\mathbf{O} \mathbf{N}) \vec u", details=False)
latex_bmatrix(O.dot(N), r"\mathbf{O} \mathbf{N}", details=False)
display(Markdown("<hr>"))
display(Markdown(r"$\mathbf{O}\mathbf{M} = \bot$"))
latex_bmatrix(O.dot(O), r"\mathbf{O} \mathbf{O}", details=False)
display(Markdown(r"$\mathbf{N}\mathbf{N} = \bot$"))
latex_bmatrix(N.dot(M), r"\mathbf{N} \mathbf{M}", details=False)
display(Markdown(r"$\mathbf{N}\mathbf{O} = \bot$"))
latex_bmatrix(M.dot(N), r"\mathbf{M} \mathbf{N}", details=False)
display(Markdown(r"$\mathbf{M}\mathbf{M} = \bot$"))
latex_bmatrix(M.dot(O), r"\mathbf{M} \mathbf{O}", details=False)
def inwendige_producten():
RNG().set(3)
random_tensor(r"\vec u", 2)
random_tensor(r"\vec v", 3)
random_tensor(r"\vec w", 2)
random_tensor(r"\vec x", 4)
random_tensor(r"\vec y", 4)
def determinanten():
RNG().set(6)
random_tensor(r"\textbf{M}", (2, 2), singular=NONDEGENERATE)
random_tensor(r"\textbf{N}", (2, 2), singular=DEGENERATE)
random_tensor(r"\textbf{O}", (2, 2), singular=NONDEGENERATE)
random_tensor(r"\textbf{P}", (3, 3),
singular=NONDEGENERATE,
interval=(0, 5))
random_tensor(r"\textbf{Q}", (3, 3), singular=DEGENERATE, interval=(0, 5))
def inwendige_producten():
# pylint: disable=C0103
RNG().set(3)
u = random_tensor(r"\vec u", 2, ret=True, details=False)
v = random_tensor(r"\vec v", 3, ret=True, details=False)
w = random_tensor(r"\vec w", 2, ret=True, details=False)
x = random_tensor(r"\vec x", 4, ret=True, details=False)
y = random_tensor(r"\vec y", 4, ret=True, details=False)
display(Markdown("<hr>"))
dot = lambda u, v: (u.T@v)[0][0]
table = f"""$\\begin{{array}}{{|r|c|c|c|c|c|}} \\hline
& \\vec u & \\vec v & \\vec w & \\vec x & \\vec y \\\\ \\hline
\\vec u & {dot(u,u)}
& \\bot
& {dot(u,w)}
& \\bot
& \\bot
\\\\ \\hline
\\vec v & \\bot
& {dot(v,v)}
& \\bot
& \\bot
& \\bot
\\\\ \\hline
\\vec w & {dot(w,u)}
& \\bot
& {dot(w,w)}
& \\bot
& \\bot
\\\\ \\hline
\\vec x & \\bot
& \\bot
& \\bot
& {dot(x,x)}
& {dot(x,y)}
\\\\ \\hline
\\vec y & \\bot
& \\bot
& \\bot
& {dot(y,x)}
& {dot(y,y)}
\\\\ \\hline
\\end{{array}}$"""
display(Markdown(table))
display(Markdown("<hr>"))
display(Markdown(r"**Wat valt je op qua symmetrie aan de tabel? Is $\langle \vec u | \vec v\rangle$ hetzelfde als $\langle \vec v | \vec u \rangle$?**"))
display(Markdown(r"*Ja, dus de volgorde maakt niet uit / het inwendig product is hier commutatief.*"))
display(Markdown(r"**Heeft $\langle \vec u | \vec u \rangle$ een speciale betekenis? Of de wortel daarvan?**"))
display(Markdown(r"*De wortel van het inwendig product van een vector met zichzelf is de lengte van de vector (stelling van Pythagoras)*"))
def inverses():
# pylint: disable=C0103
RNG().set(6)
def fr_matrix(M, divisor=1, label=None):
def fraction(n):
n = int(n)
gcd = np.gcd(n,divisor)
s = '' if divisor*n > 0 else '-'
n = abs((n/gcd).round(0).astype(int))
d = abs((divisor/gcd).round(0).astype(int))
if d == 1:
return s+str(n)
else:
return f"{s}\\frac{{{n}}}{{{d}}}"
if len(M.shape) > 2:
raise ValueError('bmatrix can at most display two dimensions')
lines = str(M).replace("[", "").replace("]", "").splitlines()
if label:
result = [label + " = "]
else:
result = [""]
result += [r"\begin{bmatrix}"]
result += [" " + " & ".join(map(fraction, l.split())) + r"\\" for l in lines]
result += [r"\end{bmatrix}"]
display(Math("\n".join(result)))
adj = lambda M: np.array(((M[1][1], -M[0][1]),(-M[1][0], M[0][0])))
M = random_tensor(r"\textbf{M}", (2,2), singular=NONDEGENERATE, ret=True, details=False)
det = np.linalg.det(M).round(0).astype(int)
latex_bmatrix(adj(M), r"$\text{adj}(\mathbf{M})", details=False)
display(Markdown(f"$\\text{{det}}(\\mathbf{{M}}) = {det}$"))
fr_matrix(adj(M), det, r"$\mathbf{M}^{-1}")
display(Markdown("<hr>"))
N = random_tensor(r"\textbf{N}", (2,2), singular=DEGENERATE, ret=True, details=False)
latex_bmatrix(adj(N), r"$\text{adj}(\mathbf{N})", details=False)
display(Markdown(f"$\\text{{det}}(\\mathbf{{N}}) = {np.linalg.det(N).round(0).astype(int)}$"))
display(Markdown(r"$\mathbf{N}^{-1} = \bot$"))
display(Markdown("<hr>"))
O = random_tensor(r"\textbf{O}", (2,2), singular=NONDEGENERATE, ret=True, details=False)
det = np.linalg.det(O).round(0).astype(int)
latex_bmatrix(adj(O), r"$\text{adj}(\mathbf{O})", details=False)
display(Markdown(f"$\\text{{det}}(\\mathbf{{O}}) = {det}$"))
fr_matrix(adj(O), det, r"$\mathbf{O}^{-1}")
def matrix_producten():
RNG().set(4)
random_tensor(r"\vec u", 3)
RNG().consume_entropy(0x02, -0x14, 0x14)
random_tensor(r"\mathbf{M}", (3, 2))
random_tensor(r"\mathbf{N}", (2, 3))
random_tensor(r"\mathbf{O}", (2, 2))
def determinanten():
# pylint: disable=C0103
RNG().set(6)
M = random_tensor(r"\textbf{M}", (2,2), singular=NONDEGENERATE, ret=True, details=False)
N = random_tensor(r"\textbf{N}", (2,2), singular=DEGENERATE, ret=True, details=False)
O = random_tensor(r"\textbf{O}", (2,2), singular=NONDEGENERATE, ret=True, details=False)
P = random_tensor(r"\textbf{P}", (3,3), singular=NONDEGENERATE, interval=(0,5), ret=True, details=False)
Q = random_tensor(r"\textbf{Q}", (3,3), singular=DEGENERATE, interval=(0,5), ret=True, details=False)
display(Markdown("<hr>"))
display(Markdown(f"$\\text{{det}}(\\mathbf{{M}}) = {np.linalg.det(M).round(0).astype(int)}$"))
display(Markdown(f"$\\text{{det}}(\\mathbf{{N}}) = {np.linalg.det(N).round(0).astype(int)}$"))
display(Markdown(f"$\\text{{det}}(\\mathbf{{O}}) = {np.linalg.det(O).round(0).astype(int)}$"))
display(Markdown(f"$\\text{{det}}(\\mathbf{{P}}) = {np.linalg.det(P).round(0).astype(int)}$"))
display(Markdown(f"$\\text{{det}}(\\mathbf{{Q}}) = {np.linalg.det(Q).round(0).astype(int)}$"))
def matrix_vector():
# pylint: disable=C0103
RNG().set(4)
u = random_tensor(r"\vec u", 3, ret=True, details=False)
v = random_tensor(r"\vec v", 2, ret=True, details=False)
M = random_tensor(r"\mathbf{M}", (3,2), ret=True, details=False)
N = random_tensor(r"\mathbf{N}", (2,3), ret=True, details=False)
O = random_tensor(r"\mathbf{O}", (2,2), ret=True, details=False)
RNG().set(2).consume_entropy(0x06, -0x14, 0x14)
pa = random_tensor(r"\vec {p_a}", 2, ret=True, details=False)
pb = random_tensor(r"\vec {p_b}", 2, ret=True, details=False)
qa = random_tensor(r"\vec {q_a}", 4, ret=True, details=False)
qb = random_tensor(r"\vec {q_b}", 4, ret=True, details=False)
display(Markdown("<hr>"))
latex_bmatrix(M.dot(v), r"\mathbf{M}\vec{v}", details=False)
display(Markdown(r"$\mathbf{M}\vec{u} = \bot$"))
display(Markdown(r"$\mathbf{N}\vec{v} = \bot$"))
latex_bmatrix(N.dot(u), r"\mathbf{N}\vec{u}", details=False)
latex_bmatrix(O.dot(N.dot(u)), r"\mathbf{O} (\mathbf{N} \vec u)", details=False)
display(Markdown("<hr>"))
P = np.hstack((pa,pb))
Q = np.hstack((qa,qb))
latex_bmatrix(P, r"\mathbf{P}", details=False)
latex_bmatrix(Q, r"\mathbf{Q}", details=False)
display(Markdown("<hr>"))
latex_bmatrix(P.dot(np.array((3,4))).reshape(2,1), r"\mathbf{P} \begin{bmatrix}3 \\ 4\end{bmatrix}", details=False)
latex_bmatrix(Q.dot(np.array((8,-0.5))).reshape(4,1), r"\mathbf{Q} \begin{bmatrix}8 \\ -\frac{1}{2}\end{bmatrix}", details=False)
def negatieven_en_sommen():
# pylint: disable=C0103
RNG().set(0)
u = random_tensor(r"\vec u", ret=True, details=False)
v = random_tensor(r"\vec v", ret=True, details=False)
w = random_tensor(r"\vec w", 3, ret=True, details=False)
x = random_tensor(r"\vec x", 3, ret=True, details=False)
y = random_tensor(r"\vec y", 5, ret=True, details=False)
z = random_tensor(r"\vec z", 5, ret=True, details=False)
display(Markdown("<hr>"))
latex_bmatrix(-u, r"-\vec u", details=False)
latex_bmatrix(-v, r"-\vec v", details=False)
latex_bmatrix(-w + x, r"-\vec w + \vec x", details=False)
latex_bmatrix(-y-z, r"- \vec y - \vec z", details=False)
def lineaire_combinaties():
# pylint: disable=C0103
RNG().set(1)
u = random_tensor(r"\vec u", ret=True, details=False)
v = random_tensor(r"\vec v", ret=True, details=False)
w = random_tensor(r"\vec w", 2, ret=True, details=False)
x = random_tensor(r"\vec x", 2, ret=True, details=False)
y = random_tensor(r"\vec y", 4, ret=True, details=False)
z = random_tensor(r"\vec z", 4, ret=True, details=False)
display(Markdown("<hr>"))
latex_bmatrix(3*u, r"3\vec{u}", details=False)
latex_bmatrix(-5*v, r"-5\vec{v}", details=False)
latex_bmatrix(v/2, r"\frac{1}{2} \vec{v}", details=False)
latex_bmatrix(3*w + 4*x, r"3\vec{w} + 4\vec{x}", details=False)
latex_bmatrix(8*y-z/2, r"8\vec{y} - \frac{1}{2}\vec{z}", details=False)
def rank():
# pylint: disable=C0103
RNG().set(7)
def stats(M, label):
def basis(vs):
res = r"\left\{"
res += ", ".join([r"\begin{bmatrix}" + r"\\".join([str(c) for c in v]) + r"\end{bmatrix}" for v in vs])
res += r"\right\}"
return res
U = lu(M)[2]
cols = []
for i in range(U.shape[0]):
r = np.flatnonzero(U[i, :])
if r.size > 0:
cols.append(r[0])
cols = dict.fromkeys(cols)
display(Markdown(f"$\\text{{I}}(\\textbf{{{label}}}) = {basis([M[:,c] for c in cols])}$"))
display(Markdown(f"$\\text{{Rank}}(\\textbf{{{label}}}) = {np.linalg.matrix_rank(M)}$"))
display(Markdown(f"$\\text{{Ker}}(\\textbf{{{label}}}) = {basis(Matrix(M).nullspace())}$"))
display(Markdown(f"$\\text{{Nullity}}(\\textbf{{{label}}}) = {M.shape[1] - np.linalg.matrix_rank(M)}$"))
M = random_tensor(r"\textbf{M}", (2,2), singular=NONDEGENERATE, interval=(0,5), ret=True, details=False)
stats(M, 'M')
display(Markdown("<hr>"))
N = random_tensor(r"\textbf{N}", (2,2), singular=DEGENERATE, interval=(0,5), ret=True, details=False)
stats(N, 'N')
display(Markdown("<hr>"))
O = random_tensor(r"\textbf{O}", (3,3), singular=NONDEGENERATE, interval=(0,5), ret=True, details=False)
stats(O, 'O')
display(Markdown("<hr>"))
P = random_tensor(r"\textbf{P}", (3,3), singular=DEGENERATE, interval=(0,5), ret=True, details=False)
stats(P, 'P')
display(Markdown("<hr>"))
Q = random_tensor(r"\textbf{Q}", (2,3), singular=DONT_CARE, interval=(0,5), ret=True, details=False)
stats(Q, 'Q')
display(Markdown("<hr>"))
R = random_tensor(r"\textbf{R}", (2,3), singular=DONT_CARE, interval=(0,5), ret=True, details=False)
stats(R, 'R')
RNG().set(1)
def rank():
RNG().set(7)
random_tensor(r"\textbf{M}", (2, 2),
singular=NONDEGENERATE,
interval=(0, 5))
random_tensor(r"\textbf{N}", (2, 2), singular=DEGENERATE, interval=(0, 5))
random_tensor(r"\textbf{O}", (3, 3),
singular=NONDEGENERATE,
interval=(0, 5))
random_tensor(r"\textbf{P}", (3, 3), singular=DEGENERATE, interval=(0, 5))
random_tensor(r"\textbf{Q}", (2, 3), singular=DONT_CARE, interval=(0, 5))
random_tensor(r"\textbf{R}", (2, 3), singular=DONT_CARE, interval=(0, 5))
def inverses():
RNG().set(6)
random_tensor(r"\textbf{M}", (2, 2), singular=NONDEGENERATE)
random_tensor(r"\textbf{N}", (2, 2), singular=DEGENERATE)
random_tensor(r"\textbf{O}", (2, 2), singular=NONDEGENERATE)
def matrix_vector():
RNG().set(4)
random_tensor(r"\vec u", 3)
random_tensor(r"\vec v", 2)
random_tensor(r"\mathbf{M}", (3, 2))
random_tensor(r"\mathbf{N}", (2, 3))
random_tensor(r"\mathbf{O}", (2, 2))
RNG().set(2).consume_entropy(0x06, -0x14, 0x14)
random_tensor(r"\vec {p_a}", 2)
random_tensor(r"\vec {p_b}", 2)
random_tensor(r"\vec {q_a}", 4)
random_tensor(r"\vec {q_b}", 4)
def lineaire_combinaties():
RNG().set(1)
random_tensor(r"\vec u")
random_tensor(r"\vec v")
RNG().set(2)
random_tensor(r"\vec w", 2)
random_tensor(r"\vec x", 2)
random_tensor(r"\vec y", 4)
random_tensor(r"\vec z", 4)
def negatieven_en_sommen():
RNG().set(0)
random_tensor(r"\vec u")
random_tensor(r"\vec v")
random_tensor(r"\vec w", 3)
random_tensor(r"\vec x", 3)
random_tensor(r"\vec y", 5)
random_tensor(r"\vec z", 5)
