Graphische beschreibung von Matrix Multiplikation

matrix/projektion.py

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+#%%
+import matplotlib.pyplot as plt
+import numpy as np
+from mpl_toolkits.mplot3d import Axes3D
+
+fig = plt.figure(figsize=(9, 7))
+ax = fig.add_subplot(111, projection='3d')
+
+# Vektoren definieren
+vektoren = {
+    r'$\vec{a}$': (2, -1, 2),
+    r'$P \cdot \vec{a}$': (2, -1, 0)
+}
+
+farben = ['#2196F3', '#FF5722']
+
+for (name, wert), farbe in zip(vektoren.items(), farben):
+    ax.quiver(0, 0, 0, *wert, color=farbe, lw=2.5, arrow_length_ratio=0.1)
+    # Beschriftung am Pfeilende
+    ax.text(wert[0] * 1.05, wert[1] * 1., wert[2] + 0.1,
+            name, fontsize=13, color=farbe, fontweight='bold')
+
+# Projektion von a auf die xy-Ebene gestrichelt
+ax.plot([2, 2], [-1, -1], [0, 2], 'k--', lw=1, alpha=0.5)
+
+# Achsen
+ax.set_xlabel('x')
+ax.set_ylabel('y')
+ax.set_zlabel('z')
+ax.set_title('Projektion', fontsize=14)
+
+# Ansichtswinkel
+ax.view_init(elev=20, azim=45)
+
+plt.tight_layout()
+plt.savefig('vektoren_3d.png', dpi=150)
+plt.show()

matrix/rotation.py

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+#%%
+import numpy as np
+import matplotlib.pyplot as plt
+from matplotlib.patches import Arc
+
+fig, ax = plt.subplots(figsize=(7, 7))
+
+# Define vectors
+a = np.array([2, 2])
+R90_a = np.array([-2, 2])
+R135_a = np.array([-2 * np.sqrt(2), 0])
+
+# Draw vectors
+vectors = [
+    (a, "steelblue", "$\\vec{a}$"),
+    (R90_a, "coral", "$R(90°)\\vec{a}$"),
+    (R135_a, "seagreen", "$R(135°)\\vec{a}$"),
+]
+
+for vec, color, label in vectors:
+    ax.annotate("", xy=vec, xytext=(0, 0),
+                arrowprops=dict(arrowstyle="->", color=color, lw=2))
+
+# Labels (individual placement for fine control)
+ax.text(a[0] * 1.1, a[1] * 1.1, "$\\vec{a}$", fontsize=12, color="steelblue",
+        ha="center", va="center")
+ax.text(R90_a[0] * 1.1, R90_a[1] * 1.1, "$R(90°)\\vec{a}$", fontsize=12, color="coral",
+        ha="center", va="center")
+ax.text(R135_a[0] - 0.4, R135_a[1], "$R(135°)\\vec{a}$", fontsize=12, color="seagreen",
+        ha="center", va="center")
+
+# Compute angles from x-axis
+angle_a = np.degrees(np.arctan2(a[1], a[0]))
+angle_R90 = np.degrees(np.arctan2(R90_a[1], R90_a[0]))
+angle_R135 = np.degrees(np.arctan2(R135_a[1], R135_a[0]))
+
+# Arc 1: angle between a and R(90°)×a
+arc1 = Arc((0, 0), 2.0, 2.0, angle=0,
+           theta1=angle_a, theta2=angle_R90,
+           color="coral", linestyle="--", lw=1.5)
+ax.add_patch(arc1)
+
+mid1 = np.radians((angle_a + angle_R90) / 2)
+ax.text(1.2 * np.cos(mid1), 1.2 * np.sin(mid1),
+        f"{angle_R90 - angle_a:.0f}°", fontsize=11, color="coral",
+        ha="center", va="center")
+
+# Arc 2: angle between a and R(135°)×a
+arc2 = Arc((0, 0), 3.2, 3.2, angle=0,
+           theta1=angle_a, theta2=angle_R135,
+           color="seagreen", linestyle="--", lw=1.5)
+ax.add_patch(arc2)
+
+mid2 = np.radians((angle_a + angle_R135) / 2)
+ax.text(1.9 * np.cos(mid2), 1.9 * np.sin(mid2),
+        f"{angle_R135 - angle_a:.0f}°", fontsize=11, color="seagreen",
+        ha="center", va="center")
+
+# Axes
+ax.axhline(0, color="lightgray", lw=0.5)
+ax.axvline(0, color="lightgray", lw=0.5)
+ax.set_xlim(-4, 3)
+ax.set_ylim(-0.5, 3)
+ax.set_aspect("equal")
+ax.grid(True, alpha=0.3)
+ax.set_title("Rotation zu $\\vec{a}$")
+plt.tight_layout()
+plt.show()
+# %%

matrix/streckung_stauchung.py

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+#%%
+import matplotlib.pyplot as plt
+import numpy as np
+
+fig, ax = plt.subplots(figsize=(8, 8))
+
+# Vektoren definieren
+vektoren = {
+    r'$\vec{a}$': (2, 1),
+    r'$S \cdot \vec{a}$': (1, 2),
+    r'$S \cdot (S \cdot \vec{a})$': (0.5, 4)
+}
+
+farben = ['#2196F3', '#FF5722', '#4CAF50']
+
+for (name, wert), farbe in zip(vektoren.items(), farben):
+    ax.annotate('', xy=wert, xytext=(0, 0),
+                arrowprops=dict(arrowstyle='->', lw=2.5, color=farbe))
+    ax.text(wert[0] * 0.5 + 0.2, wert[1] * 0.5 + 0.2, name,
+            fontsize=13, color=farbe, fontweight='bold')
+
+# Achsen und Gitter
+ax.axhline(0, color='grey', lw=0.5)
+ax.axvline(0, color='grey', lw=0.5)
+ax.set_xlim(-0.5, 3)
+ax.set_ylim(-0.5, 4.5)
+ax.set_aspect('equal')
+ax.grid(True, alpha=0.3)
+ax.set_title('Strecken / Stauchung', fontsize=14)
+
+plt.tight_layout()
+plt.savefig('vektoren.png', dpi=150)
+plt.show()
+# %%