2017-02-01.sagews

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:

• The week 4 feedback survey is still open: https://docs.google.com/forms/d/e/1FAIpQLScbt2Lg-zgz6kDyDSvB0t8jXX2FDPQJdFxdqpOsX4VrNwuvZw/viewform

• 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.

• References for today's material:

  • Linear algebra quickref: https://wiki.sagemath.org/quickref?action=AttachFile&do=get&target=quickref-linalg.pdf

  • Linear algebra reference manual: http://doc.sagemath.org/html/en/reference/index.html#linear-algebra

  • 1.5, 4.4, 4.16, of Sage For Undergraduates: http://www.gregorybard.com/Sage.html

• Warning: Linear algebra code in Sage is a mix of very fast functionality written by researchers, and some VERY slow functionality added for educational purposes in made by teachers, with no cleer warning either way. Also functionality that was written by researchers that used to be fast, but was "accidentally" made slow by somebody later.

# 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)

$\displaystyle \left(\begin{array}{rrr} 1 & 2 & 3 \\ \sqrt{2} & 5 & \pi \end{array}\right)$

v = vector([1, sqrt(2), pi^2]); v

(1, sqrt(2), pi^2)

show(v)

$\displaystyle \left(1,\,\sqrt{2},\,\pi^{2}\right)$

show(A); show(v); show(A*v)

$\displaystyle \left(\begin{array}{rrr} 1 & 2 & 3 \\ \sqrt{2} & 5 & \pi \end{array}\right)$

$\displaystyle \left(1,\,\sqrt{2},\,\pi^{2}\right)$

$\displaystyle \left(3 \, \pi^{2} + 2 \, \sqrt{2} + 1,\,\pi^{3} + 6 \, \sqrt{2}\right)$

v * A.transpose()

(3*pi^2 + 2*sqrt(2) + 1, pi^3 + 6*sqrt(2))

show(A.echelon_form())

$\displaystyle \left(\begin{array}{rrr} 1 & 0 & \frac{2 \, {\left(\pi - 3 \, \sqrt{2}\right)}}{2 \, \sqrt{2} - 5} + 3 \\ 0 & 1 & -\frac{\pi - 3 \, \sqrt{2}}{2 \, \sqrt{2} - 5} \end{array}\right)$

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)

$\displaystyle \left(\begin{array}{rrrrr} 1 & 2 & -\frac{1}{2} & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 \\ -\frac{1}{2} & 0 & 2 & 0 & 2 \\ -1 & 2 & -1 & 0 & 2 \\ 1 & 0 & 2 & 2 & 0 \end{array}\right)$

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)

$\displaystyle \left(\begin{array}{rrrrr} 1 & 2 & -\frac{1}{2} & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 \\ -\frac{1}{2} & 0 & 2 & 0 & 2 \\ -1 & 2 & -1 & 0 & 2 \\ 1 & 0 & 2 & 2 & 0 \end{array}\right)$

$\displaystyle \left(\begin{array}{rrrrr} 2 & 2 & -\frac{1}{2} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ -\frac{1}{2} & 0 & 3 & 0 & 2 \\ -1 & 2 & -1 & 1 & 2 \\ 1 & 0 & 2 & 2 & 1 \end{array}\right)$

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())

$\displaystyle (x - 0.714003959174589 - 2.07933489614716i) \cdot (x - 0.714003959174589 + 2.07933489614716i) \cdot (x + 1.28534665449870 - 1.03106393820580i) \cdot (x + 1.28534665449870 + 1.03106393820580i) \cdot (x + 1.85731460935178)$

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)

$\displaystyle \left(\begin{array}{rrrrr} 1 & 2 & -\frac{1}{2} & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 \\ -\frac{1}{2} & 0 & 2 & 0 & 2 \\ -1 & 2 & -1 & 0 & 2 \\ 1 & 0 & 2 & 2 & 0 \end{array}\right)$

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)

$\displaystyle \left(1,\,0,\,1,\,1,\,0\right)$ $\displaystyle \left(1,\,0,\,1,\,1,\,0\right)$

A = matrix([[1,1],[2,3]])

show(A)

$\displaystyle \left(\begin{array}{rr} 1 & 1 \\ 2 & 3 \end{array}\right)$

v = vector([1,-1])

show(A, v)

$\displaystyle \left(\begin{array}{rr} 1 & 1 \\ 2 & 3 \end{array}\right)$ $\displaystyle \left(1,\,-1\right)$

w = A.solve_right(v); w

(4, -3)

show(w*A, v)

$\displaystyle \left(1,\,-1\right)$ $\displaystyle \left(1,\,-1\right)$