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Foundations of coding: theory and applications of error-correcting codes

Jiri Adamek

Chapter 7

Linear Algebra - all with Video Answers

Educators

Chapter Questions

Problem 1

Prove that multiplication in every ring has a unique neutral element.

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04:04

Problem 2

Verify that in every field, $\left(x^{-1}\right)^{-1}=x$ for all $x \neq 0$.

Elham Kordzadeh
Elham Kordzadeh
Numerade Educator

Problem 3

Prove that in every ring, all elements $x$, for which $x^{-1}$ exists, form a group under multiplication. Apply this to $\mathbf{Z}_{12}$.

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02:00

Problem 4

Write down the operation tables of the field $\mathrm{Z}_5$. Find $x^{-1}$ for each $x \neq 0$.

James Chok
James Chok
Numerade Educator

Problem 5

Is $\mathrm{Z}_4$ a field? Can you re-define the multiplication (leaving the addition) to obtain a field?

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02:28

Problem 6

Verify that the following tables
$$
\begin{array}{c|cccc}
+ & 0 & 1 & p & q \\
\hline 0 & 0 & 1 & p & q \\
1 & 1 & 0 & q & p \\
p & p & q & 0 & 1 \\
q & q & p & 1 & 0
\end{array}
$$
$$
\begin{array}{c|llll}
\cdot & 0 & 1 & p & q \\
\hline 0 & 0 & 0 & 0 & 0 \\
1 & 0 & 1 & p & q \\
p & 0 & p & q & 1 \\
q & 0 & q & 1 & p
\end{array}
$$
define a four-element field $F=\{0,1, p, q\}$.

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator

Problem 7

How many vectors are there in the linear space $\mathbf{Z}_2^n$ ? In $\mathbf{Z}_3^n$ ?

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Problem 8

Verify that the linear space $F^n$ over an $r$-element field $F$ has precisely
(1) $r^n-1$ linear subspaces of dimension 1. [Hint: the number of all vectors, including 0 , is $r^n$.]
(2) $r^n-1$ linear subspaces of dimension $n-1$. [Hint: apply (1) to the orthogonal complement.]

Victor Salazar
Victor Salazar
Numerade Educator
15:19

Problem 9

Prove that the set of all $n \times m$ matrices with the addition and scalar multiplication of 7.6 form a linear space. What is its dimension?

Anthony Ramos
Anthony Ramos
Numerade Educator
03:19

Problem 10

Prove that the linear space of Example (4) in 7.4 is not finite-dimensional.

Anthony Ramos
Anthony Ramos
Numerade Educator
02:54

Problem 11

Find a row-echelon form of the following matrix
$$
\left[\begin{array}{lllllll}
1 & 0 & 1 & 0 & 1 & 0 & 1 \\
1 & 1 & 0 & 0 & 1 & 1 & 0 \\
0 & 1 & 1 & 0 & 0 & 1 & 1
\end{array}\right]
$$
(1) over the field $\mathrm{Z}_2$,
(2) over the field $\mathrm{Z}_3$.
In both cases, determine the rank.

AG
Ankit Gupta
Numerade Educator
02:09

Problem 12

Find all solutions of the following system of equations over $\mathrm{Z}_2$ and over $\mathbf{Z}_3$ :
$$
\begin{aligned}
& x_1+x_2+x_4=0, \\
& x_3+x_4+x_5=0, \\
& x_1+x_3+x_5=0 .
\end{aligned}
$$

Dale Sanford
Dale Sanford
Numerade Educator
09:11

Problem 13

Find an inverse matrix to the following matrix
$$
\left[\begin{array}{lll}
1 & 0 & 1 \\
1 & 1 & 1 \\
0 & 1 & 0
\end{array}\right]
$$
(1) over the field $Z_2$,
(2) over the field $\mathbf{Z}_3$.

Vishvajeetkumar Bhaskar Batule
Vishvajeetkumar Bhaskar Batule
Numerade Educator
09:11

Problem 14

Find an inverse matrix of the following matrix
$$
A=\left[\begin{array}{llll}
1 & 0 & 1 & 1 \\
1 & 1 & 0 & 1 \\
1 & 1 & 1 & 0 \\
0 & 0 & 1 & 1
\end{array}\right]
$$
(1) over the field $Z_2$,
(2) over the field $\mathrm{Z}_3$.
In both cases, discuss the solutions of the system of equation $\mathbf{A x} \mathbf{x}^{\mathrm{tr}}=\mathbf{b}^{\mathbf{t r}}$, where $\mathbf{b}=\mathbf{1 1 0 0}$.

Vishvajeetkumar Bhaskar Batule
Vishvajeetkumar Bhaskar Batule
Numerade Educator
05:26

Problem 15

Given a solution $\mathbf{a}=a_1 \ldots a_n$ of a nonhomogenous system of linear equations $\mathbf{A x}^{\mathrm{tr}}=\mathrm{d}^{\mathrm{tr}}$ (see Remark 7.6), prove that all solutions of that system have the form $a+b$, where $b$ solves the corresponding homogenous system (i.e., $\mathbf{A b}^{\mathrm{tr}}=\mathbf{0}^{\mathrm{tr}}$ ). In other words, the collection of all solutions is the coset (6.2)
$$
\mathrm{a}+K^{\perp}
$$
modulo the linear space $K^{\perp}$ which is the orthogonal complement of the row space $K$ of the matrix $\mathbf{A}$.

Jack Chen
Jack Chen
Numerade Educator
06:19

Problem 16

Find all solutions of the following system of equations over $\mathbf{Z}_5$ :
$$
\begin{aligned}
& x_1+3 x_2+2 x_3=1, \\
& x_1+2 x_2+x_3=2 .
\end{aligned}
$$

Bobby Barnes
Bobby Barnes
University of North Texas

Problem 17

Find the orthogonal complement of the Hamming code of length 7 (see 5.5).

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Problem 18

What property of the field $F$ of real numbers guarantees that for each linear subspace $L$ of $F^m$, one has $L \cap L^{\perp}=\{0\}$ ? Does any of the fields $\mathbf{Z}_p$ have this property? Do you know any other field which does?

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Problem 19

Let $A$ be an $m \times n$ matrix of rank $m$. Prove the following:
(1) A row-echelon form of $\mathbf{A}$ has 1's in the main diagonal $\left(a_{i i}=1\right)$ and zeros under it.
(2) If $m=n$, then it is possible to find elementary row operations which transform $\mathbf{A}$ into an identity matrix.

Victor Salazar
Victor Salazar
Numerade Educator

Problem 20

Prove the following generalization of Theorem 7.2: each number $i=$ $0,1, \ldots, n-1$ relatively prime with $n$ has an inverse element in the ring $\mathbf{Z}_n$; i.e., there exists $j$ with $i j \equiv 1(\bmod n)$.

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Problem 21

Prove that if a matrix $\mathbf{A}$ is obtained by concatenating matrices $\mathbf{A}_1$ and $\mathbf{A}_2$ row by row (notation: $\mathbf{A}=\left[\mathbf{A}_1 \mid \mathbf{A}_2\right]$ ) and a matrix $\mathbf{B}$ is obtained by concatenating matrices $\mathbf{B}_1$ and $\mathbf{B}_2$ column by column (notation: $\mathbf{B}=\left[\frac{\mathbf{B}_1}{\mathbf{B}_2}\right]$ ), then for the matrix product, the following formula holds:
$$
A B=\left[A_1 \mid A_2\right]\left[\frac{B_1}{B_2}\right]=A_1 B_1+A_2 B_2,
$$
whenever the number of rows of $\mathbf{A}_i$ is equal to the number of columns of $\mathbf{B}_i$ for $i=1,2$.

Victor Salazar
Victor Salazar
Numerade Educator