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 random_tensor(label=None, size=None, singular=DONT_CARE,\
interval=None, ret=False, details=True):
# pylint: disable=R0913
def generate_tensor(size, interval):
if not interval:
interval = (-20, 20)
if size and isinstance(size, int):
size = (size, 1)
elif not size or not isinstance(size, tuple):
size = (np.random.randint(2, 6), 1)
return np.random.randint(interval[0], interval[1], size=size)
candidate = generate_tensor(size, interval)
while (singular == DEGENERATE and np.linalg.det(candidate) != 0)\
or (singular == NONDEGENERATE and np.linalg.det(candidate) == 0):
candidate = generate_tensor(size, interval)
latex_bmatrix(candidate, label, details=details)
return candidate if ret else None
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 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"$\mathbf{O}^{-1}")
def matrix_producten():
# pylint: disable=C0103
RNG().set(4)
u = random_tensor(r"\vec u", 3, ret=True, details=False)
RNG().consume_entropy(0x02, -0x14, 0x14)
M = random_tensor(r"\mathbf{M}", (3,2), ret=True, details=False)
N = random_tensor(r"\mathbf{N}", (2,3), ret=True, details=False)
O = random_tensor(r"\mathbf{O}", (2,2), ret=True, details=False)
display(Markdown("<hr>"))
latex_bmatrix(O.dot(N.dot(u)), r"\mathbf{O} (\mathbf{N} \vec u)", details=False)
latex_bmatrix(O.dot(N).dot(u), r"(\mathbf{O} \mathbf{N}) \vec u", details=False)
latex_bmatrix(O.dot(N), r"\mathbf{O} \mathbf{N}", details=False)
display(Markdown("<hr>"))
display(Markdown(r"$\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$"))
latex_bmatrix(M.dot(O), r"\mathbf{M} \mathbf{O}", details=False)
def matrix_vector():
# pylint: disable=C0103
RNG().set(4)
u = random_tensor(r"\vec u", 3, ret=True, details=False)
v = random_tensor(r"\vec v", 2, ret=True, details=False)
M = random_tensor(r"\mathbf{M}", (3,2), ret=True, details=False)
N = random_tensor(r"\mathbf{N}", (2,3), ret=True, details=False)
O = random_tensor(r"\mathbf{O}", (2,2), ret=True, details=False)
RNG().set(2).consume_entropy(0x06, -0x14, 0x14)
pa = random_tensor(r"\vec {p_a}", 2, ret=True, details=False)
pb = random_tensor(r"\vec {p_b}", 2, ret=True, details=False)
qa = random_tensor(r"\vec {q_a}", 4, ret=True, details=False)
qb = random_tensor(r"\vec {q_b}", 4, ret=True, details=False)
display(Markdown("<hr>"))
latex_bmatrix(M.dot(v), r"\mathbf{M}\vec{v}", details=False)
display(Markdown(r"$\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)