Linear and Geometric Algebra

Walk through theorems and exercises in the book.



In [1]:

    
from sympy import *
from galgebra.printer import Format
Format()
from galgebra.ga import Ga

Chapter 1: Vectors



In [2]:

    
(x, y, z) = xyz = symbols('x,y,z',real=True)
(o3d, ex, ey, ez) = Ga.build('e_x e_y e_z', g=[1, 1, 1], coords=xyz)



In [3]:

    
V = o3d



In [4]:

    
a = V.mv('a', 'scalar')
b = V.mv('b', 'scalar')
c = V.mv('c', 'scalar')
d = V.mv('d', 'scalar')
u = V.mv('u', 'vector')
v = V.mv('v', 'vector')
w = V.mv('w', 'vector')



In [5]:

    
def vector(ga, components):
    bases = ga.mv()
    return sum([components[i] * e for i, e in enumerate(bases)])

Theorem 1.8



In [6]:

    
v + w == w + v









    Out[6]:





True



In [7]:

    
(u + v) + w == u + (v + w)









    Out[7]:





True



In [8]:

    
v + S(0) == v









    Out[8]:





True



In [9]:

    
S(0) * v == S(0)









    Out[9]:





True



In [10]:

    
S(1) * v == v









    Out[10]:





True



In [11]:

    
a * (b * v) == (a * b) * v









    Out[11]:





True



In [12]:

    
a * (v + w) == a * v + a * w









    Out[12]:





True



In [13]:

    
(a + b) * v ==  a * v + b * v









    Out[13]:





True

Problem 1.1.4



In [14]:

    
uu = vector(V, [1, 2, 3])
vv = vector(V, [4, 5, 6])
ww = vector(V, [5, 6, 7])



In [15]:

    
uu + vv









    Out[15]:





\begin{equation*} 5 \boldsymbol{e}_{x} + 7 \boldsymbol{e}_{y} + 9 \boldsymbol{e}_{z} \end{equation*}



In [16]:

    
7 * uu + 2 * ww









    Out[16]:





\begin{equation*} 17 \boldsymbol{e}_{x} + 26 \boldsymbol{e}_{y} + 35 \boldsymbol{e}_{z} \end{equation*}



In [17]:

    
7 * uu - 2 * ww









    Out[17]:





\begin{equation*} -3 \boldsymbol{e}_{x} + 2 \boldsymbol{e}_{y} + 7 \boldsymbol{e}_{z} \end{equation*}



In [18]:

    
3 * uu + 2 * vv + ww









    Out[18]:





\begin{equation*} 16 \boldsymbol{e}_{x} + 22 \boldsymbol{e}_{y} + 28 \boldsymbol{e}_{z} \end{equation*}

Problems 1.2



In [19]:

    
v + S(-1) * v









    Out[19]:





\begin{equation*}  0  \end{equation*}

Chapter 2: Vector Spaces

Exercise 2.7



In [20]:

    
v0 = vector(V, [0, 0, 0])



In [21]:

    
a * v0 == v0









    Out[21]:





True

Problem 2.1.5



In [22]:

    
(-a) * v == a * (-v)









    Out[22]:





True



In [23]:

    
(-a) * v == - a * v









    Out[23]:





True



In [24]:

    
sqrt(2) * u + Rational(2, 3) * v









    Out[24]:





\begin{equation*} \left ( \sqrt{2} u^{x} + \frac{2 v^{x}}{3}\right ) \boldsymbol{e}_{x} + \left ( \sqrt{2} u^{y} + \frac{2 v^{y}}{3}\right ) \boldsymbol{e}_{y} + \left ( \sqrt{2} u^{z} + \frac{2 v^{z}}{3}\right ) \boldsymbol{e}_{z} \end{equation*}

Chapter 3 Matrices



In [25]:

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









    Out[25]:





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



In [26]:

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









    Out[26]:





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



In [27]:

    
MAB = MA * MB
MAB









    Out[27]:





$$\left[\begin{matrix}5 & 4\\5 & -13\end{matrix}\right]$$



In [28]:

    
MAB_00 = MA[0, :] * MB[:, 0]
MAB_00









    Out[28]:





$$\left[\begin{matrix}5\end{matrix}\right]$$



In [29]:

    
MAB.inv()









    Out[29]:





$$\left[\begin{matrix}\frac{13}{85} & \frac{4}{85}\\\frac{1}{17} & - \frac{1}{17}\end{matrix}\right]$$



In [30]:

    
MAB.inv() * MAB.det()









    Out[30]:





$$\left[\begin{matrix}-13 & -4\\-5 & 5\end{matrix}\right]$$



In [31]:

    
MAB.inv() * MAB









    Out[31]:





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

Exercise 3.11



In [32]:

    
a, b, c, d = symbols('a b c d')
M = Matrix([[a, b], [c, d]])
M









    Out[32]:





$$\left[\begin{matrix}a & b\\c & d\end{matrix}\right]$$



In [33]:

    
M.det()









    Out[33]:





$$a d - b c$$



In [34]:

    
M.inv()









    Out[34]:





$$\left[\begin{matrix}\frac{d}{a d - b c} & - \frac{b}{a d - b c}\\- \frac{c}{a d - b c} & \frac{a}{a d - b c}\end{matrix}\right]$$



In [35]:

    
Mul(1 / M.det(), M.inv() * M.det())









    Out[35]:





$$\frac{\left[\begin{matrix}d & - b\\- c & a\end{matrix}\right]}{a d - b c}$$

Theorem 3.5



In [36]:

    
a, b, c, d, e, f, g, h = symbols('a b c d e f g h')
A = Matrix([[a, b], [c, d]])
B =Matrix([[e, f], [g, h]])



In [37]:

    
A*B









    Out[37]:





$$\left[\begin{matrix}a e + b g & a f + b h\\c e + d g & c f + d h\end{matrix}\right]$$



In [38]:

    
(A * B).inv()









    Out[38]:





$$\left[\begin{matrix}\frac{\left(a e + b g\right) \left(c f + d h\right) - \left(a f + b h\right) \left(- c e - d g\right) - \left(a f + b h\right) \left(c e + d g\right)}{\left(a e + b g\right) \left(\left(a e + b g\right) \left(c f + d h\right) - \left(a f + b h\right) \left(c e + d g\right)\right)} & - \frac{a f + b h}{\left(a e + b g\right) \left(c f + d h\right) - \left(a f + b h\right) \left(c e + d g\right)}\\\frac{- c e - d g}{\left(a e + b g\right) \left(c f + d h\right) - \left(a f + b h\right) \left(c e + d g\right)} & \frac{a e + b g}{\left(a e + b g\right) \left(c f + d h\right) - \left(a f + b h\right) \left(c e + d g\right)}\end{matrix}\right]$$



In [39]:

    
B.inv() * A.inv()









    Out[39]:





$$\left[\begin{matrix}\frac{c f}{\left(a d - b c\right) \left(e h - f g\right)} + \frac{d h}{\left(a d - b c\right) \left(e h - f g\right)} & - \frac{a f}{\left(a d - b c\right) \left(e h - f g\right)} - \frac{b h}{\left(a d - b c\right) \left(e h - f g\right)}\\- \frac{c e}{\left(a d - b c\right) \left(e h - f g\right)} - \frac{d g}{\left(a d - b c\right) \left(e h - f g\right)} & \frac{a e}{\left(a d - b c\right) \left(e h - f g\right)} + \frac{b g}{\left(a d - b c\right) \left(e h - f g\right)}\end{matrix}\right]$$



In [40]:

    
simplify((A * B).inv()) == simplify(B.inv() * A.inv())









    Out[40]:





False



In [41]:

    
simplify((A * B).inv() - B.inv() * A.inv())









    Out[41]:





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

Corollary 3.8



In [42]:

    
A.inv().inv() == A









    Out[42]:





False



In [43]:

    
simplify(A.inv().inv() - A)









    Out[43]:





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

Exercise 3.19



In [44]:

    
A = Matrix([
    [1, -4],
    [1, -1]
])
b = Matrix([3, -1])



In [45]:

    
A









    Out[45]:





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



In [46]:

    
b









    Out[46]:





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



In [47]:

    
A.LUsolve(b)









    Out[47]:





$$\left[\begin{matrix}- \frac{7}{3}\\- \frac{4}{3}\end{matrix}\right]$$

Chapter 4 Inner Product Spaces

Theorem 4 .2



In [48]:

    
v.project_in_blade(u)









    Out[48]:





\begin{equation*} \frac{u^{x} \left(u^{x} v^{x} + u^{y} v^{y} + u^{z} v^{z}\right)}{{\left ( u^{x} \right )}^{2} + {\left ( u^{y} \right )}^{2} + {\left ( u^{z} \right )}^{2}} \boldsymbol{e}_{x} + \frac{u^{y} \left(u^{x} v^{x} + u^{y} v^{y} + u^{z} v^{z}\right)}{{\left ( u^{x} \right )}^{2} + {\left ( u^{y} \right )}^{2} + {\left ( u^{z} \right )}^{2}} \boldsymbol{e}_{y} + \frac{u^{z} \left(u^{x} v^{x} + u^{y} v^{y} + u^{z} v^{z}\right)}{{\left ( u^{x} \right )}^{2} + {\left ( u^{y} \right )}^{2} + {\left ( u^{z} \right )}^{2}} \boldsymbol{e}_{z} \end{equation*}



In [49]:

    
(u|v)/u**2*u









    Out[49]:





\begin{equation*} \frac{u^{x} \left(u^{x} v^{x} + u^{y} v^{y} + u^{z} v^{z}\right)}{{\left ( u^{x} \right )}^{2} + {\left ( u^{y} \right )}^{2} + {\left ( u^{z} \right )}^{2}} \boldsymbol{e}_{x} + \frac{u^{y} \left(u^{x} v^{x} + u^{y} v^{y} + u^{z} v^{z}\right)}{{\left ( u^{x} \right )}^{2} + {\left ( u^{y} \right )}^{2} + {\left ( u^{z} \right )}^{2}} \boldsymbol{e}_{y} + \frac{u^{z} \left(u^{x} v^{x} + u^{y} v^{y} + u^{z} v^{z}\right)}{{\left ( u^{x} \right )}^{2} + {\left ( u^{y} \right )}^{2} + {\left ( u^{z} \right )}^{2}} \boldsymbol{e}_{z} \end{equation*}



In [50]:

    
v.project_in_blade(u) == (u|v)/u









    Out[50]:





True



In [51]:

    
v.project_in_blade(w) + u.project_in_blade(w) == (u + v).project_in_blade(w)









    Out[51]:





True

Theorem 4.3



In [52]:

    
(a * u) | v == a * (u | v)









    Out[52]:





True



In [53]:

    
(u + v) | w == (u | w) + (v | w)









    Out[53]:





True



In [54]:

    
u | v == v | u









    Out[54]:





True

Definition 4 .11



In [55]:

    
v | v == v.norm2()









    Out[55]:





True

Exercise 4 .16

parallelogram identity



In [56]:

    
(u + v).norm2() + (u - v).norm2() == 2 * u.norm2() + 2 * v.norm2()









    Out[56]:





True

Lemma 4.13 (Cauchy-Schwarz inequality)



In [57]:

    
(u | v).norm() <= u.norm() * v.norm()









    Out[57]:





$$\left(u^{x} v^{x} + u^{y} v^{y} + u^{z} v^{z}\right)^{1.0} \leq \sqrt{\left(u^{x}\right)^{2} + \left(u^{y}\right)^{2} + \left(u^{z}\right)^{2}} \sqrt{\left(v^{x}\right)^{2} + \left(v^{y}\right)^{2} + \left(v^{z}\right)^{2}}$$



In [58]:

    
simplify(u.norm() * v.norm() - (u | v).norm())









    Out[58]:





$$\sqrt{\left(u^{x}\right)^{2} + \left(u^{y}\right)^{2} + \left(u^{z}\right)^{2}} \sqrt{\left(v^{x}\right)^{2} + \left(v^{y}\right)^{2} + \left(v^{z}\right)^{2}} - \left(u^{x} v^{x} + u^{y} v^{y} + u^{z} v^{z}\right)^{1.0}$$

Definition 4.15 (Angle between vectors)



In [59]:

    
def cosv(u, v):
    return (u | v) / (u.norm() * v.norm())



In [60]:

    
cosv(u, v)









    Out[60]:





\begin{equation*} \frac{u^{x} v^{x} + u^{y} v^{y} + u^{z} v^{z}}{\sqrt{{\left ( u^{x} \right )}^{2} + {\left ( u^{y} \right )}^{2} + {\left ( u^{z} \right )}^{2}} \sqrt{{\left ( v^{x} \right )}^{2} + {\left ( v^{y} \right )}^{2} + {\left ( v^{z} \right )}^{2}}} \end{equation*}

Exercise 4.32



In [67]:

    
VO4 = Ga('e', g=[1, 1, 1, 1], coords=symbols('v1,v2,v3,v4',real=True))



In [68]:

    
vo4 = vector(VO4, [1, 1, 1, 1])



In [69]:

    
vo4









    Out[69]:





\begin{equation*}  \boldsymbol{e}_{v1} + \boldsymbol{e}_{v2} + \boldsymbol{e}_{v3} + \boldsymbol{e}_{v4} \end{equation*}



In [75]:

    
e_u1 = vector(VO4, [3, 0, 4, 0])
e_u2 = vector(VO4, [0, 3, 0, 4])



In [76]:

    
e_u1









    Out[76]:





\begin{equation*} 3 \boldsymbol{e}_{v1} + 4 \boldsymbol{e}_{v3} \end{equation*}



In [77]:

    
e_u2









    Out[77]:





\begin{equation*} 3 \boldsymbol{e}_{v2} + 4 \boldsymbol{e}_{v4} \end{equation*}



In [78]:

    
u1, u2 = symbols('u1,u2',real=True)



In [91]:

    
coord_fun = u1 * e_u1 + u2 * e_u2
coord_fun









    Out[91]:





\begin{equation*} 3 u_{1} \boldsymbol{e}_{v1} + 3 u_{2} \boldsymbol{e}_{v2} + 4 u_{1} \boldsymbol{e}_{v3} + 4 u_{2} \boldsymbol{e}_{v4} \end{equation*}



In [115]:

    
coord_map = [(coord_fun | e).obj for e in VO4.mv()]
coord_map









    Out[115]:





$$\left [ 3 u_{1}, \quad 3 u_{2}, \quad 4 u_{1}, \quad 4 u_{2}\right ]$$



In [116]:

    
UO2 = VO4.sm(coord_map, [u1, u2])
UO2









    Out[116]:





<galgebra.ga.Sm at 0x10ebe9d30>



In [122]:

    
UO2.u









    Out[122]:





$$\left [ 3 u_{1}, \quad 3 u_{2}, \quad 4 u_{1}, \quad 4 u_{2}\right ]$$



In [117]:

    
UO2.g









    Out[117]:





$$\left[\begin{matrix}25 & 0\\0 & 25\end{matrix}\right]$$



In [127]:

    
UO2E = e_u1 ^ e_u2
UO2E









    Out[127]:





\begin{equation*} 9 \boldsymbol{e}_{v1}\wedge \boldsymbol{e}_{v2} + 12 \boldsymbol{e}_{v1}\wedge \boldsymbol{e}_{v4} -12 \boldsymbol{e}_{v2}\wedge \boldsymbol{e}_{v3} + 16 \boldsymbol{e}_{v3}\wedge \boldsymbol{e}_{v4} \end{equation*}



In [124]:

    
vo4.reflect_in_blade(UO2E)









    Out[124]:





\begin{equation*} \frac{17}{25} \boldsymbol{e}_{v1} + \frac{17}{25} \boldsymbol{e}_{v2} + \frac{31}{25} \boldsymbol{e}_{v3} + \frac{31}{25} \boldsymbol{e}_{v4} \end{equation*}



In [ ]: