Math 416, Abstract linear algebra (北美程序代写,美国程序代写,University of Illinois Urbana-Champaign代写,数学Math代写,Math416代写)

北美程序代写 美国程序代写 University of Illinois Urbana-Champaign代写 数学Math代写 Math416代写

For each of the following lists of vectors in R3, determine whether the first vector can be expressed as a linear combination of the other two.

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Homework 2

Math 416, Abstract linear algebra, Fall 2019 Instructor: Daesung Kim

Due date: September 13, 2019

Textbooks: In the assignment, the two texts are abbreviated as follows:

  • [FIS]: Freidberg, Insel, and Spence, Linear Algebra, 4th edition, 2002.
  • [Bee]: Beezer, A First Course in Linear Algebra, Version 3.5, 2015.

 

  1. For each of the following lists of vectors in R3, determine whether the first vector can be expressed as a linear combination of the other two.

(a) (−2, 0, 3), (1, 3, 0), (2, 4, −1)

(b) (3, 4, 1), (1, −2, 1), (−2, −1, 1)

(c) (5, 1, −5), (1, −2, −3), (−2, 3, −4)

  1.  

∈
Let V  be the set of all A         M2×2(R) such that Aij = 0 if i > j. Note that V is a subspace of M2×2(R). Let

S = 1 0, 0 1, 0 0.

 

0 0

0 0

0 1
Prove that S generates V .
  1. Let V be a vector space and S1, S2 subsets of V such that S1 ⊆ S2.
    1. Show that Span(S1) ⊆ Span(S2).
    2. Show that if S1 generates V , then S2 also generates V .
  2. Let V be a vector space and S1, S2 subsets of V . (Note that if S = ∅, then we define Span(S) = {0}.)
    1. Prove that Span(S1 ∩ S2) ⊆ Span(S1) ∩ Span(S2).
    2. Give an example in which Span(S1 ∩ S2) = Span(S1) ∩ Span(S2).
    3. Give an example in which Span(S1 ∩ S2) ∕= Span(S1) ∩ Span(S2).
  3. Consider a system of linear equations

 

 
a11x1 + a12x2 + · · · + a1nxn = b1

 

..
(∗) a21x1 + a22x2 + · · · + a2nxn = b2

 

 
am1x1 + am2x2 + · · · + amnxn = bm

where aij, bi ∈ R for i ∈ {1, 2, · · · , m}, j ∈ {1, 2, · · · , n}. Suppose that there are two distinct solutions (x1, · · · , xn) and (y1, · · · , yn). Show that there are infinitely many solutions to (∗).

  1. Let M, N, L ∈ Mm×n(R). If M is row-equivalent to N , then we denote by M ∼ N . Prove the following.
    1. M ∼ M .
    2. If M ∼ N , then N ∼ M .

1

 

Math 416                                                                                           HW 2                                                                           Due Sep 13, 2019

    1. If M ∼ N and N ∼ L, then M ∼ L.
  2. Find reduced row-echelon forms of the following matrices.

 

                              1   4   −3   −3   6(a)

1  2  −1   1       5

2  3  −1 4       8

 

                          

1   2  2   0        2

 

                           −

(b)        1   0  8   5     6

1   1  5   5        3

 

2  1    1         8

1  2        6       −1

 

1  3  10      −5

(c)  3   1   −1   15

  1. Consider a system of linear equations

 

 

..

a11x1 + a12x2 + · · · + a1nxn = 0 (∗) a21x1 + a22x2 + · · · + a2nxn = 0

 

 

am1x1 + am2x2 + · · · + amnxn = 0

where aij ∈ R for i ∈ {1, 2, · · · , m}, j ∈ {1, 2, · · · , n}. Let V be the set of all solutions (x1, x2, · · · , xn) ∈

Rn to (∗). Show that V is a subspace of Rn.

 

 

 

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