Course source code: https://codistwa.com/guides/linear-algebra. More courses on https://codistwa.com

linear-algebra.py

# ============================================================
# What is a vector?
# ============================================================

scalar = [3] # scalar
row = np.array([2,1], dtype=object) # row matrice / vector
col = np.array([ [3], [-1] ], dtype=object) # column matrice / vector

display(Math(sym.latex(sym.sympify(scalar))))
display(Math(sym.latex(sym.sympify(row))))
display(Math(sym.latex(sym.sympify(col))))

ax = plt.axes()
ax.arrow(0, 0, row[0], row[1], head_width=0.2, label='$row$', head_length=0.2, color='r', length_includes_head=True)
ax.arrow(0, 0, col[0][0], col[1][0], head_width=0.2, label='$column$', head_length=0.2, color='b', length_includes_head=True)

plt.axis('square')
plt.legend()

plt.grid()
plt.show()

plt.title('Row and columns vectors',fontsize=12)
plt.savefig('row_col_vectors.png', bbox_inches='tight')

# ============================================================
# What is a 3D vector?
# ============================================================

vector1_3D = np.array([ [2], [3], [-2] ], dtype=object)
vector2_3D = np.array([ [2], [-4], [1] ], dtype=object)
vector3_3D = np.array([ [2], [1], [4] ], dtype=object)

display(Math(sym.latex(sym.sympify(vector1_3D))))
display(Math(sym.latex(sym.sympify(vector2_3D))))
display(Math(sym.latex(sym.sympify(vector3_3D))))

fig = plt.figure()
ax = fig.add_subplot(projection='3d')

# draw vectors
ax.plot([0,vector1_3D[0]],[0,vector1_3D[1]],[0,vector1_3D[2]],'b',linewidth=3)
ax.plot([0,vector2_3D[0]],[0,vector2_3D[1]],[0,vector2_3D[2]],'r',linewidth=3)
ax.plot([0,vector3_3D[0]],[0,vector3_3D[1]],[0,vector3_3D[2]],'g',linewidth=3)

# guidelines
ax.plot([-5,5],[0,0],[0,0],'--',color=[.7,.7,.7])
ax.plot([0,0],[-5,5],[0,0],'--',color=[.7,.7,.7])
ax.plot([0,0],[0,0],[-5,5],'--',color=[.7,.7,.7])

ax.set_xlim3d(-5,5)
ax.set_ylim3d(-5,5)
ax.set_zlim3d(-5,5)

ax.set_xlabel('X')
ax.set_ylabel('Y')
ax.set_zlabel('Z')

fig.savefig('3d_vectors.png', bbox_inches='tight')

plt.show()

# ============================================================
# Sum of two vectors
# ============================================================

vector1 = np.array([ [3], [-2] ], dtype=object)
vector2 = np.array([ [2], [-4] ], dtype=object)

display(Math(sym.latex(sym.sympify(vector1))))
display(Math(sym.latex(sym.sympify(vector2))))

vector_sum = vector1 + vector2
vector_sum = np.add(vector1, vector2)

vector1_show = sym.latex(sym.sympify(vector1))
vector2_show = sym.latex(sym.sympify(vector2))
vector3_show = sym.latex(sym.sympify(vector_sum))

display(Math('%s + %s=%s' %(vector1_show, vector2_show, vector3_show)))

ax = plt.axes()
ax.arrow(0, 0, vector1[0][0], vector1[1][0], label='$vector1$', head_width=0.2, head_length=0.2, color='b', length_includes_head=True)
ax.arrow(vector1[0][0], vector1[1][0], vector2[0][0], vector2[1][0],  label='$vector2$', head_width=0.2, head_length=0.2, color='r', length_includes_head=True)
ax.arrow(0, 0, vector_sum[0][0], vector_sum[1][0], label='$vector1 + vector2$', head_width=0.2, head_length=0.2, color='g', length_includes_head=True)

plt.axis('on')
plt.legend()
plt.grid()

plt.title('Sum of two vectors',fontsize=12)
plt.savefig('sum_two_vectors.png', bbox_inches='tight')

plt.show()

# ============================================================
# Difference of two vectors
# ============================================================

vector1 = np.array([ [3], [-2] ], dtype=object)
vector2 = np.array([ [2], [-4] ], dtype=object)

vector2_inverse = np.array([ [-2], [4] ], dtype=object)

vector_difference = vector1 - vector2
vector4_show = sym.latex(sym.sympify(vector_difference))

display(Math('%s - %s=%s' %(vector1_show, vector2_show, vector4_show)))

ax = plt.axes()
ax.arrow(0, 0, vector1[0][0], vector1[1][0], head_width=0.2, label='$vector1$', head_length=0.2, color='b', length_includes_head=True)
ax.arrow(vector1[0][0], vector1[1][0], vector2_inverse[0][0], vector2_inverse[1][0], label='$vector2$', head_width=0.2, head_length=0.2, color='r', length_includes_head=True)
ax.arrow(0, 0, vector_difference[0][0], vector_difference[1][0], label='$vector1-vector2$', head_width=0.2, head_length=0.2, color='g', length_includes_head=True)

plt.title('Difference of two vectors',fontsize=12)

plt.axis('on')
plt.legend()

plt.grid()
plt.show()

plt.savefig('difference_two_vectors.png', bbox_inches='tight')

# ============================================================
# Multiplying a vector by a scalar
# ============================================================

vector_scalar = vector1 * scalar
vector5_show = sym.latex(sym.sympify(vector_scalar))

display(Math('%s X % s=%s' %(vector1_show, scalar[0], vector5_show)))

# ============================================================
# Multiplying a vector by a vector - the dot product
# ============================================================

vector_dot = vector1 * vector2

v1 = np.array([1, 2, 3])
v2 = np.array([-3, 1, -4])

print(v1 @ v2) # python operator https://www.python.org/dev/peps/pep-0465/
print(np.dot(v1, v2))