Chapter 3 problems



In [1]:

    
from sympy import *
init_printing()



In [ ]:



In [2]:

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









    Out[2]:





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



In [3]:

    
b = Matrix([6,5])



In [4]:

    
AUG = A.row_join(b)
AUG # the augmented matrix









    Out[4]:





$$\left[\begin{matrix}3 & 3 & 6\\2 & \frac{3}{2} & 5\end{matrix}\right]$$

Alice



In [5]:

    
AUGA = AUG.copy()
AUGA[0,:] = AUGA[0,:]/3
AUGA









    Out[5]:





$$\left[\begin{matrix}1 & 1 & 2\\2 & \frac{3}{2} & 5\end{matrix}\right]$$



In [6]:

    
AUGA[1,:] = AUGA[1,:] - 2*AUGA[0,:]
AUGA









    Out[6]:





$$\left[\begin{matrix}1 & 1 & 2\\0 & - \frac{1}{2} & 1\end{matrix}\right]$$



In [7]:

    
AUGA[1,:] = -2*AUGA[1,:]
AUGA









    Out[7]:





$$\left[\begin{matrix}1 & 1 & 2\\0 & 1 & -2\end{matrix}\right]$$



In [8]:

    
AUGA[0,:] = AUGA[0,:] - AUGA[1,:]
AUGA









    Out[8]:





$$\left[\begin{matrix}1 & 0 & 4\\0 & 1 & -2\end{matrix}\right]$$

Bob



In [9]:

    
AUGB = AUG.copy()
AUGB[0,:] = AUGB[0,:] - AUGB[1,:]
AUGB









    Out[9]:





$$\left[\begin{matrix}1 & \frac{3}{2} & 1\\2 & \frac{3}{2} & 5\end{matrix}\right]$$



In [10]:

    
AUGB[1,:] = AUGB[1,:] - 2*AUGB[0,:]
AUGB









    Out[10]:





$$\left[\begin{matrix}1 & \frac{3}{2} & 1\\0 & - \frac{3}{2} & 3\end{matrix}\right]$$



In [11]:

    
AUGB[1,:] = -1*S(2)/3*AUGB[1,:]
AUGB









    Out[11]:





$$\left[\begin{matrix}1 & \frac{3}{2} & 1\\0 & 1 & -2\end{matrix}\right]$$



In [12]:

    
AUGB[0,:] = AUGB[0,:] - S(3)/2*AUGB[1,:]
AUGB









    Out[12]:





$$\left[\begin{matrix}1 & 0 & 4\\0 & 1 & -2\end{matrix}\right]$$



In [ ]:

Charlotte



In [13]:

    
AUGC = AUG.copy()
AUGC[0,:], AUGC[1,:] = AUGC[1,:], AUGC[0,:]
AUGC









    Out[13]:





$$\left[\begin{matrix}2 & \frac{3}{2} & 5\\3 & 3 & 6\end{matrix}\right]$$



In [14]:

    
AUGC[0,:] = AUGC[0,:]/2
AUGC









    Out[14]:





$$\left[\begin{matrix}1 & \frac{3}{4} & \frac{5}{2}\\3 & 3 & 6\end{matrix}\right]$$



In [15]:

    
AUGC[1,:] = AUGC[1,:] - 3*AUGC[0,:]
AUGC









    Out[15]:





$$\left[\begin{matrix}1 & \frac{3}{4} & \frac{5}{2}\\0 & \frac{3}{4} & - \frac{3}{2}\end{matrix}\right]$$



In [16]:

    
AUGC[1,:] = S(4)/3*AUGC[1,:]
AUGC









    Out[16]:





$$\left[\begin{matrix}1 & \frac{3}{4} & \frac{5}{2}\\0 & 1 & -2\end{matrix}\right]$$



In [17]:

    
AUGC[0,:] = AUGC[0,:] - S(3)/4*AUGC[1,:]
AUGC









    Out[17]:





$$\left[\begin{matrix}1 & 0 & 4\\0 & 1 & -2\end{matrix}\right]$$

P3.3



In [18]:

    
# define agmented matrices for three systems of eqns. with unique sol'ns
A = Matrix([
        [ -1, -2, -2],
        [  3, 3, 0]])
        
B = Matrix([
        [ 1, -1, -2,  1],
        [-2,  3,  3, -1],
        [-1,  0,  1,  2]])

C = Matrix([
        [ 2, -2,  3, 2],
        [ 1, -2, -1, 0],
        [-2,  2,  2, 1]])



In [19]:

    
A









    Out[19]:





$$\left[\begin{matrix}-1 & -2 & -2\\3 & 3 & 0\end{matrix}\right]$$



In [20]:

    
A.rref()









    Out[20]:





$$\left ( \left[\begin{matrix}1 & 0 & -2\\0 & 1 & 2\end{matrix}\right], \quad \left ( 0, \quad 1\right )\right )$$



In [21]:

    
B









    Out[21]:





$$\left[\begin{matrix}1 & -1 & -2 & 1\\-2 & 3 & 3 & -1\\-1 & 0 & 1 & 2\end{matrix}\right]$$



In [22]:

    
B.rref()









    Out[22]:





$$\left ( \left[\begin{matrix}1 & 0 & 0 & -4\\0 & 1 & 0 & -1\\0 & 0 & 1 & -2\end{matrix}\right], \quad \left ( 0, \quad 1, \quad 2\right )\right )$$



In [23]:

    
C









    Out[23]:





$$\left[\begin{matrix}2 & -2 & 3 & 2\\1 & -2 & -1 & 0\\-2 & 2 & 2 & 1\end{matrix}\right]$$



In [24]:

    
C.rref()









    Out[24]:





$$\left ( \left[\begin{matrix}1 & 0 & 0 & - \frac{2}{5}\\0 & 1 & 0 & - \frac{1}{2}\\0 & 0 & 1 & \frac{3}{5}\end{matrix}\right], \quad \left ( 0, \quad 1, \quad 2\right )\right )$$

P3.4



In [25]:

    
# now for three systems of eqns. with infinitely many sol'ns
D = Matrix([
        [ -1, -2, -2],
        [  3, 6,   6]])
        
E = Matrix([
        [ 1, -1, -2,  1],
        [-2,  3,  3, -1],
        [-1,  2,  1,  0]])

F = Matrix([
        [ 2, -2, 3, 2],
        [ 0,  0, 5, 3],
        [-2,  2, 2, 1]])

Solving d)



In [26]:

    
D









    Out[26]:





$$\left[\begin{matrix}-1 & -2 & -2\\3 & 6 & 6\end{matrix}\right]$$



In [27]:

    
D.rref()









    Out[27]:





$$\left ( \left[\begin{matrix}1 & 2 & 2\\0 & 0 & 0\end{matrix}\right], \quad \left ( 0\right )\right )$$



In [28]:

    
D[0:2,0:2].nullspace()









    Out[28]:





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



In [29]:

    
# the solutions to the sytem of equations represented by D
# is of the form    point + nullspace
point = D.rref()[0][:,2]
nullspace = D[0:2,0:2].nullspace()



In [30]:

    
# the point is also called he particular solution
point









    Out[30]:





$$\left[\begin{matrix}2\\0\end{matrix}\right]$$



In [31]:

    
# if A aug matrix is [A|b], then the point satisfies A*point = b.
print( D[0:2,0:2]*point == D[:,2] )
D[0:2,0:2]*point









    



True






    Out[31]:





$$\left[\begin{matrix}-2\\6\end{matrix}\right]$$

Null space



In [32]:

    
# the nullspace of A in aug. matrix [A|b] is one dimensional and spanned by
n = nullspace[0]
n
# every vector n in the nullspace of A satisfies  A*n=0









    Out[32]:





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



In [33]:

    
# so solution to A*x=b is any (point+s*n) where s is any real number
# since  A*(point +s*n) = A*point + sA*n = A*point + 0 = b.
# verify claim for 20 values of s in range -5,-4,-3,-2,-1,0,1,2,3,4,5
for s in range(-5,6):
    print( D[0:2,0:2]*(point + s*n), 
           D[0:2,0:2]*(point + s*n) == D[:,2] )









    



Matrix([[-2], [6]]) True
Matrix([[-2], [6]]) True
Matrix([[-2], [6]]) True
Matrix([[-2], [6]]) True
Matrix([[-2], [6]]) True
Matrix([[-2], [6]]) True
Matrix([[-2], [6]]) True
Matrix([[-2], [6]]) True
Matrix([[-2], [6]]) True
Matrix([[-2], [6]]) True
Matrix([[-2], [6]]) True

Solving e)



In [34]:

    
E









    Out[34]:





$$\left[\begin{matrix}1 & -1 & -2 & 1\\-2 & 3 & 3 & -1\\-1 & 2 & 1 & 0\end{matrix}\right]$$



In [35]:

    
E.rref()









    Out[35]:





$$\left ( \left[\begin{matrix}1 & 0 & -3 & 2\\0 & 1 & -1 & 1\\0 & 0 & 0 & 0\end{matrix}\right], \quad \left ( 0, \quad 1\right )\right )$$



In [36]:

    
point_E = E.rref()[0][:,3]
nullspace_E = E[0:3,0:3].nullspace()[0]
s = symbols('s')
point_E + s*nullspace_E









    Out[36]:





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

Solving f)



In [37]:

    
F









    Out[37]:





$$\left[\begin{matrix}2 & -2 & 3 & 2\\0 & 0 & 5 & 3\\-2 & 2 & 2 & 1\end{matrix}\right]$$



In [38]:

    
F.rref()









    Out[38]:





$$\left ( \left[\begin{matrix}1 & -1 & 0 & \frac{1}{10}\\0 & 0 & 1 & \frac{3}{5}\\0 & 0 & 0 & 0\end{matrix}\right], \quad \left ( 0, \quad 2\right )\right )$$



In [39]:

    
point_F = F.rref()[0][:,3]
nullspace_F = F[0:3,0:3].nullspace()[0]
s = symbols('s')
point_F + s*nullspace_F









    Out[39]:





$$\left[\begin{matrix}s + \frac{1}{10}\\s + \frac{3}{5}\\0\end{matrix}\right]$$



In [ ]: