

In [1]:

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

Linearity



In [2]:

    
b, m = symbols('b m')

def f(x):
    return m*x



In [3]:

    
f(1)









    Out[3]:





$$m$$



In [4]:

    
f(2)









    Out[4]:





$$2 m$$



In [5]:

    
f(1+2)









    Out[5]:





$$3 m$$



In [6]:

    
f(1) + f(2)









    Out[6]:





$$3 m$$



In [7]:

    
expand(f(x+y)) ==  f(x) + f(y)









    Out[7]:





True

What about vector inputs?



In [8]:

    
m_1, m_2 = symbols('m_1 m_2')

def T(vec):
    """A function that takes a 2D vector and returns a number."""
    return m_1*vec[0] + m_2*vec[1]



In [9]:

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



In [10]:

    
T(u)









    Out[10]:





$$m_{1} u_{1} + m_{2} u_{2}$$



In [11]:

    
T(v)









    Out[11]:





$$m_{1} v_{1} + m_{2} v_{2}$$



In [12]:

    
T(u) + T(v)









    Out[12]:





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



In [13]:

    
expand( T(u+v) )









    Out[13]:





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



In [14]:

    
simplify( T(alpha*u + beta*v) - alpha*T(u) - beta*T(v) )









    Out[14]:





$$0$$

Linear transformations

A linear transformation is function that takes vectors as inputs, and produces vectors as outputs:

$$ T: \mathbb{R}^n \to \mathbb{R}^m. $$

see page 116 in book



In [15]:

    
m_11, m_12, m_21, m_22 = symbols('m_11 m_12 m_21 m_22')

def T(vec):
    """A linear transformations R^2 --> R^2."""
    out_1 = m_11*vec[0] + m_12*vec[1]
    out_2 = m_21*vec[0] + m_22*vec[1]
    return Matrix([out_1, out_2])



In [16]:

    
T(u)









    Out[16]:





$$\left[\begin{matrix}m_{11} u_{1} + m_{12} u_{2}\\m_{21} u_{1} + m_{22} u_{2}\end{matrix}\right]$$



In [17]:

    
T(v)









    Out[17]:





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



In [18]:

    
T(u+v)









    Out[18]:





$$\left[\begin{matrix}m_{11} \left(u_{1} + v_{1}\right) + m_{12} \left(u_{2} + v_{2}\right)\\m_{21} \left(u_{1} + v_{1}\right) + m_{22} \left(u_{2} + v_{2}\right)\end{matrix}\right]$$

Linear transformations as matrix-vector products

see page 113



In [19]:

    
def T_impl(vec):
    """A linear transformations implemented as matrix-vector product."""
    M_T = Matrix([[m_11, m_12], 
                  [m_21, m_22]])
    return M_T*vec



In [20]:

    
T_impl(u)









    Out[20]:





$$\left[\begin{matrix}m_{11} u_{1} + m_{12} u_{2}\\m_{21} u_{1} + m_{22} u_{2}\end{matrix}\right]$$



In [ ]: