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Math 152: Intro to Mathematical Software

2017-02-01

Kiran Kedlaya; University of California, San Diego

adapted from lectures by William Stein, University of Washington

Lecture 10: Linear Algebra (part 1)


%md
Announcements:

- The week 4 feedback survey is still open: (URL removed)
- The "solutions" for HW 3, i.e., the original TeX files that I used to make the answer samples, are available as a new handout.
- If you think you are on the waitlist and did *not* receive an email yesterday asking you to reconfirm your interest in enrolling in the class, please let me know before I leave class today.

Announcements:

# You should read this
matrix?
# and this
vector?

matrix.block







#A = matrix(2, 3, [[1,2,3], [sqrt(2),5,pi]])
A = matrix(2, 3, [1,2,3,  sqrt(2),5,pi])
A
[ 1 2 3] [sqrt(2) 5 pi] 
A = matrix([[1,2,3],[sqrt(2), 5, pi]])

A.transpose()
[ 1 sqrt(2)] [ 2 5] [ 3 pi] 
show(A)
(12325π)\displaystyle \left(\begin{array}{rrr} 1 & 2 & 3 \\ \sqrt{2} & 5 & \pi \end{array}\right)
v = vector([1, sqrt(2), pi^2]); v
(1, sqrt(2), pi^2) 
show(v)
(1, 2, π2)\displaystyle \left(1,\,\sqrt{2},\,\pi^{2}\right)
show(A); show(v); show(A*v)
(12325π)\displaystyle \left(\begin{array}{rrr} 1 & 2 & 3 \\ \sqrt{2} & 5 & \pi \end{array}\right)
(1, 2, π2)\displaystyle \left(1,\,\sqrt{2},\,\pi^{2}\right)
(3 π2+2 2+1, π3+6 2)\displaystyle \left(3 \, \pi^{2} + 2 \, \sqrt{2} + 1,\,\pi^{3} + 6 \, \sqrt{2}\right)
v * A.transpose()
(3*pi^2 + 2*sqrt(2) + 1, pi^3 + 6*sqrt(2)) 




show(A.echelon_form())
(102 (π−3 2)2 2−5+301−π−3 22 2−5)\displaystyle \left(\begin{array}{rrr} 1 & 0 & \frac{2 \, {\left(\pi - 3 \, \sqrt{2}\right)}}{2 \, \sqrt{2} - 5} + 3 \\ 0 & 1 & -\frac{\pi - 3 \, \sqrt{2}}{2 \, \sqrt{2} - 5} \end{array}\right)

2. Standard functions:

  • determinant

  • characteristic polynomial

  • echelon form

  • row space

  • column space

  • rows, columns

  • matrix_from_rows, matrix_from_columns

  • solve

Integers() is ZZ
True 
#set_random_seed(28)  # put your age here
A = random_matrix(QQ, 5)
show(A)
(12−12000−1000−120202−12−10210220)\displaystyle \left(\begin{array}{rrrrr} 1 & 2 & -\frac{1}{2} & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 \\ -\frac{1}{2} & 0 & 2 & 0 & 2 \\ -1 & 2 & -1 & 0 & 2 \\ 1 & 0 & 2 & 2 & 0 \end{array}\right)
f = A.charpoly()
print(f.factor())
parent(f)
(x + 1) * (x^4 - 3*x^3 - 25/4*x^2 + 21*x - 13) Univariate Polynomial Ring in x over Rational Field 
B = A +1
show(A)
show(B)
(12−12000−1000−120202−12−10210220)\displaystyle \left(\begin{array}{rrrrr} 1 & 2 & -\frac{1}{2} & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 \\ -\frac{1}{2} & 0 & 2 & 0 & 2 \\ -1 & 2 & -1 & 0 & 2 \\ 1 & 0 & 2 & 2 & 0 \end{array}\right)
(22−120000000−120302−12−11210221)\displaystyle \left(\begin{array}{rrrrr} 2 & 2 & -\frac{1}{2} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ -\frac{1}{2} & 0 & 3 & 0 & 2 \\ -1 & 2 & -1 & 1 & 2 \\ 1 & 0 & 2 & 2 & 1 \end{array}\right)
B.charpoly().factor()
B.det()
x * (x^4 - 7*x^3 + 35/4*x^2 + 41/2*x - 145/4) 0 
f.base_ring()
Rational Field 
f.factor()
x^5 + 3*x^4 + 6*x^3 + 63/4*x^2 + 29*x + 195/8 
show(f.change_ring(CC).factor())
(x−0.714003959174589−2.07933489614716i)⋅(x−0.714003959174589+2.07933489614716i)⋅(x+1.28534665449870−1.03106393820580i)⋅(x+1.28534665449870+1.03106393820580i)⋅(x+1.85731460935178)\displaystyle (x - 0.714003959174589 - 2.07933489614716i) \cdot (x - 0.714003959174589 + 2.07933489614716i) \cdot (x + 1.28534665449870 - 1.03106393820580i) \cdot (x + 1.28534665449870 + 1.03106393820580i) \cdot (x + 1.85731460935178)
E= A.eigenspaces_right()

E[0]act_on_polynomial
(-1.857314609351780?, Vector space of degree 5 and dimension 1 over Algebraic Field User basis matrix: [ 1 3.504212994256084? -0.10512329983419517? 0.3761439167989345? -3.684097802778675?]) 
len(E)
5 
A.determinant()
13 
B.echelon_form()
[ 1 0 0 0 -2/3] [ 0 1 0 0 29/36] [ 0 0 1 0 5/9] [ 0 0 0 1 5/18] [ 0 0 0 0 0] 

B.row_space()
Vector space of degree 5 and dimension 4 over Rational Field Basis matrix: [ 1 0 0 0 -2/3] [ 0 1 0 0 29/36] [ 0 0 1 0 5/9] [ 0 0 0 1 5/18] 
QQ^3
Vector space of dimension 3 over Rational Field 
span(QQ, [vector([1,2,3]), vector([4,5,6]), vector([7,8,9])])
Vector space of degree 3 and dimension 2 over Rational Field Basis matrix: [ 1 0 -1] [ 0 1 2] 
show(A)
(12−12000−1000−120202−12−10210220)\displaystyle \left(\begin{array}{rrrrr} 1 & 2 & -\frac{1}{2} & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 \\ -\frac{1}{2} & 0 & 2 & 0 & 2 \\ -1 & 2 & -1 & 0 & 2 \\ 1 & 0 & 2 & 2 & 0 \end{array}\right)
v = vector([1,0,1,1,0])

w = A.solve_right(v); print w
(12/13, 0, -2/13, -4/13, 23/26) 
show(A*w, v)
(1, 0, 1, 1, 0)\displaystyle \left(1,\,0,\,1,\,1,\,0\right) (1, 0, 1, 1, 0)\displaystyle \left(1,\,0,\,1,\,1,\,0\right)
A = matrix([[1,1],[2,3]])

show(A)
(1123)\displaystyle \left(\begin{array}{rr} 1 & 1 \\ 2 & 3 \end{array}\right)
v = vector([1,-1])

show(A, v)
(1123)\displaystyle \left(\begin{array}{rr} 1 & 1 \\ 2 & 3 \end{array}\right) (1, −1)\displaystyle \left(1,\,-1\right)
w = A.solve_right(v); w
(4, -3) 
show(w*A, v)
(1, −1)\displaystyle \left(1,\,-1\right) (1, −1)\displaystyle \left(1,\,-1\right)