In [1]:

    

# setup SymPy
from sympy import *
x, y, z, t = symbols('x y z t')
init_printing()

# setup plotting
%matplotlib notebook
import matplotlib.pyplot as mpl
from util.plot_helpers import plot_vec, plot_vecs, autoscale_arrows


    

In [2]:

    

v = Matrix([1,2,3])
v


    

    
Out[2]:





$$\left[\begin{matrix}1\\2\\3\end{matrix}\right]$$

In [3]:

    

# define symbolically
v_1, v_2, v_3 = symbols('v_1 v_2 v_3')
v = Matrix([v_1,v_2,v_3])


    

In [4]:

    

v


    

    
Out[4]:





$$\left[\begin{matrix}v_{1}\\v_{2}\\v_{3}\end{matrix}\right]$$

In [5]:

    

v.T


    

    
Out[5]:





$$\left[\begin{matrix}v_{1} & v_{2} & v_{3}\end{matrix}\right]$$

In [6]:

    

A = Matrix(
    [   [1,7],
        [2,8], 
        [3,9]   ])
A


    

    
Out[6]:





$$\left[\begin{matrix}1 & 7\\2 & 8\\3 & 9\end{matrix}\right]$$

In [7]:

    

# define symbolically
a_11, a_12, a_21, a_22, a_31, a_32 = symbols('a_11 a_12 a_21 a_22 a_31 a_32')
A = Matrix([
        [a_11, a_12],
        [a_21, a_22], 
        [a_31, a_32]])


    

In [8]:

    

A


    

    
Out[8]:





$$\left[\begin{matrix}a_{11} & a_{12}\\a_{21} & a_{22}\\a_{31} & a_{32}\end{matrix}\right]$$

In [9]:

    

u_1, u_2, u_3 = symbols('u_1 u_2 u_3')
u = Matrix([u_1,u_2,u_3])
v_1, v_2, v_3 = symbols('v_1 v_2 v_3')
v = Matrix([v_1,v_2,v_3])
alpha = symbols('alpha')

u


    

    
Out[9]:





$$\left[\begin{matrix}u_{1}\\u_{2}\\u_{3}\end{matrix}\right]$$

In [10]:

    

alpha*u


    

    
Out[10]:





$$\left[\begin{matrix}\alpha u_{1}\\\alpha u_{2}\\\alpha u_{3}\end{matrix}\right]$$

In [11]:

    

u+v


    

    
Out[11]:





$$\left[\begin{matrix}u_{1} + v_{1}\\u_{2} + v_{2}\\u_{3} + v_{3}\end{matrix}\right]$$

In [12]:

    

u.norm()


    

    
Out[12]:





$$\sqrt{\left|{u_{1}}\right|^{2} + \left|{u_{2}}\right|^{2} + \left|{u_{3}}\right|^{2}}$$

In [13]:

    

uhat = u/u.norm()
uhat


    

    
Out[13]:





$$\left[\begin{matrix}\frac{u_{1}}{\sqrt{\left|{u_{1}}\right|^{2} + \left|{u_{2}}\right|^{2} + \left|{u_{3}}\right|^{2}}}\\\frac{u_{2}}{\sqrt{\left|{u_{1}}\right|^{2} + \left|{u_{2}}\right|^{2} + \left|{u_{3}}\right|^{2}}}\\\frac{u_{3}}{\sqrt{\left|{u_{1}}\right|^{2} + \left|{u_{2}}\right|^{2} + \left|{u_{3}}\right|^{2}}}\end{matrix}\right]$$

In [14]:

    

u = Matrix([1.5,1])
w = 2*u
uhat = u/u.norm()

fig = mpl.figure()
plot_vecs(u, w, uhat)
autoscale_arrows()


    

In [15]:

    

u = Matrix([u_1,u_2,u_3])
v = Matrix([v_1,v_2,v_3])

u.dot(v)


    

    
Out[15]:





$$u_{1} v_{1} + u_{2} v_{2} + u_{3} v_{3}$$

In [16]:

    

fig = mpl.figure()
u = Matrix([1,1])
v = Matrix([3,0])
plot_vecs(u,v)
autoscale_arrows()

u_dot_v = u.dot(v)
u_dot_v


    

    















    












    
Out[16]:





$$3$$

In [17]:

    

phi = acos( u.dot(v)/(u.norm()*v.norm()) )
print('angle between u and v is', phi)
u.norm()*v.norm()*cos(phi)


    

    




angle between u and v is pi/4






    
Out[17]:





$$3$$

In [18]:

    

u = Matrix([u_1,u_2,u_3])
v = Matrix([v_1,v_2,v_3])

u.cross(v)


    

    
Out[18]:





$$\left[\begin{matrix}u_{2} v_{3} - u_{3} v_{2}\\- u_{1} v_{3} + u_{3} v_{1}\\u_{1} v_{2} - u_{2} v_{1}\end{matrix}\right]$$

In [19]:

    

u = Matrix([1,0,0])
v = Matrix([1,1,0])
w = u.cross(v)      # a vector perpendicular to both u and v

mpl.figure()
plot_vecs(u, v, u.cross(v))


    

In [20]:

    

print('length of cross product', w.norm())

phi = acos( u.dot(v)/(u.norm()*v.norm()) )

w.norm() == u.norm()*v.norm()*sin(phi)


    

    




length of cross product 1






    
Out[20]:





True

In [21]:

    

def proj(vec, d):
    """Computes the projection of vector `vec` onto vector `d`."""
    return d.dot(vec)/d.norm() * d/d.norm()


    

In [22]:

    

fig = mpl.figure()
u = Matrix([1,1])
v = Matrix([3,0])

pu_on_v = proj(u,v)

plot_vecs(u, v, pu_on_v)


# autoscale_arrows()
ax = mpl.gca()
ax.set_xlim([-1,3])
ax.set_ylim([-1,3])


    

    















    












    
Out[22]:





$$\left ( -1, \quad 3\right )$$

In [23]:

    

a_11, a_12, a_21, a_22, a_31, a_32 = symbols('a_11 a_12 a_21 a_22 a_31 a_32')
A = Matrix([
        [a_11, a_12],
        [a_21, a_22], 
        [a_31, a_32]])
b_11, b_12, b_21, b_22, b_31, b_32 = symbols('b_11 b_12 b_21 b_22 b_31 b_32')
B = Matrix([
        [b_11, b_12],
        [b_21, b_22], 
        [b_31, b_32]])
alpha = symbols('alpha')


    

In [24]:

    

A


    

    
Out[24]:





$$\left[\begin{matrix}a_{11} & a_{12}\\a_{21} & a_{22}\\a_{31} & a_{32}\end{matrix}\right]$$

In [25]:

    

A + B


    

    
Out[25]:





$$\left[\begin{matrix}a_{11} + b_{11} & a_{12} + b_{12}\\a_{21} + b_{21} & a_{22} + b_{22}\\a_{31} + b_{31} & a_{32} + b_{32}\end{matrix}\right]$$

In [26]:

    

alpha*A


    

    
Out[26]:





$$\left[\begin{matrix}a_{11} \alpha & a_{12} \alpha\\a_{21} \alpha & a_{22} \alpha\\a_{31} \alpha & a_{32} \alpha\end{matrix}\right]$$

In [27]:

    

v_1, v_2 = symbols('v_1 v_2')
v = Matrix([v_1,v_2])

A*v


    

    
Out[27]:





$$\left[\begin{matrix}a_{11} v_{1} + a_{12} v_{2}\\a_{21} v_{1} + a_{22} v_{2}\\a_{31} v_{1} + a_{32} v_{2}\end{matrix}\right]$$

In [28]:

    

A[:,0]*v[0] + A[:,1]*v[1]


    

    
Out[28]:





$$\left[\begin{matrix}a_{11} v_{1} + a_{12} v_{2}\\a_{21} v_{1} + a_{22} v_{2}\\a_{31} v_{1} + a_{32} v_{2}\end{matrix}\right]$$

In [29]:

    

A = Matrix([
        [a_11,a_12],
        [a_21, a_22], 
        [a_31, a_32]])
B = Matrix([
        [b_11,b_12],
        [b_21, b_22]])

A*B


    

    
Out[29]:





$$\left[\begin{matrix}a_{11} b_{11} + a_{12} b_{21} & a_{11} b_{12} + a_{12} b_{22}\\a_{21} b_{11} + a_{22} b_{21} & a_{21} b_{12} + a_{22} b_{22}\\a_{31} b_{11} + a_{32} b_{21} & a_{31} b_{12} + a_{32} b_{22}\end{matrix}\right]$$

In [30]:

    

A.T


    

    
Out[30]:





$$\left[\begin{matrix}a_{11} & a_{21} & a_{31}\\a_{12} & a_{22} & a_{32}\end{matrix}\right]$$

In [31]:

    

print('the shape of v is ', v.shape)
v


    

    




the shape of v is  (2, 1)






    
Out[31]:





$$\left[\begin{matrix}v_{1}\\v_{2}\end{matrix}\right]$$

In [32]:

    

print('the shape of v.T is ', v.T.shape)
v.T


    

    




the shape of v.T is  (1, 2)






    
Out[32]:





$$\left[\begin{matrix}v_{1} & v_{2}\end{matrix}\right]$$

In [33]:

    

u = Matrix([u_1,u_2,u_3])
v = Matrix([v_1,v_2,v_3])

u.T*v


    

    
Out[33]:





$$\left[\begin{matrix}u_{1} v_{1} + u_{2} v_{2} + u_{3} v_{3}\end{matrix}\right]$$

In [34]:

    

u * v.T


    

    
Out[34]:





$$\left[\begin{matrix}u_{1} v_{1} & u_{1} v_{2} & u_{1} v_{3}\\u_{2} v_{1} & u_{2} v_{2} & u_{2} v_{3}\\u_{3} v_{1} & u_{3} v_{2} & u_{3} v_{3}\end{matrix}\right]$$

In [35]:

    

A = Matrix([
  [3,       3],
  [2,  S(3)/2]
])
A


    

    
Out[35]:





$$\left[\begin{matrix}3 & 3\\2 & \frac{3}{2}\end{matrix}\right]$$

In [36]:

    

A.inv()


    

    
Out[36]:





$$\left[\begin{matrix}-1 & 2\\\frac{4}{3} & -2\end{matrix}\right]$$

In [37]:

    

A * A.inv()


    

    
Out[37]:





$$\left[\begin{matrix}1 & 0\\0 & 1\end{matrix}\right]$$

In [38]:

    

A.inv() * A


    

    
Out[38]:





$$\left[\begin{matrix}1 & 0\\0 & 1\end{matrix}\right]$$

In [39]:

    

B = Matrix([
        [b_11,b_12],
        [b_21, b_22]])
B


    

    
Out[39]:





$$\left[\begin{matrix}b_{11} & b_{12}\\b_{21} & b_{22}\end{matrix}\right]$$

In [40]:

    

B.trace()


    

    
Out[40]:





$$b_{11} + b_{22}$$

In [41]:

    

B.det()


    

    
Out[41]:





$$b_{11} b_{22} - b_{12} b_{21}$$

In [ ]: