Review of Linear Algebra

Abstract

In this chapter, we will review the fundamental concepts of linear algebra. We will give the definitions of group, field, vector space, basis, dimension, vector subspace, etc. Linear block codes are vector subspaces, and to understand the concept of subspaces, you need to know groups and fields. For this reason, to understand the concept of channel code construction, it is very critical to have sufficient knowledge of linear algebra. Especially, the topic of vector spaces should be comprehended very well for the understanding of linear block codes. Nonlinear codes are also available in the literature; however, they are not used in practical systems although they exist. Literature focuses on the design of linear codes rather than nonlinear codes. In this book, we will mainly study binary linear block codes. For this reason, our focus in this chapter will be on the vector spaces constructed with the elements of binary field.

Problems

• 1. The finite set G and the operation ⊕ are given as

$$ G=\left\{0,1,2,3\right\} $$

$$ \oplus \to \mathrm{Mod}-4\ \mathrm{addititon}\ \mathrm{operation}. $$

• Determine whether G is a group under the defined operation ⊕ or not.

• 2. The finite set G and the operation ⊗ are given as

$$ G=\left\{0,1,2,3,4\right\} $$

$$ \otimes \to \mathrm{Mod}-5\ \mathrm{multiplication}\ \mathrm{operation}. $$

• Determine whether G is a group under the defined operation ⊗ or not.

• 3. The finite set G and the operation ⊗ are given as

$$ G=\left\{1,2,3,4\right\} $$

$$ \otimes \to \mathrm{Mod}-5\ \mathrm{multiplication}\ \mathrm{operation}. $$

• Show that G is a group under Mod-5 multiplication operation. Find a subgroup of G, and determine all the cosets using the subgroup.

• 4. The set of integers Z under addition operation + is a group. A subgroup of Z can be formed as

  $$ H=6Z\to H=\left\{\dots -12,-6,0,6,12,\dots \right\}. $$

• Find the cosets of H.

• 5. The finite set S and the operations ⊕ and ⊗ are given as

$$ S=\left\{0,1,2,3,4,5,6,7,8,9,10\right\} $$

$$ \oplus \to \mathrm{Mod}-11\ \mathrm{addition}\ \mathrm{operation} $$

$$ \otimes \to \mathrm{Mod}-11\ \mathrm{multiplication}\ \mathrm{operation}. $$

• Determine whether S is a field or not under the defined operations ⊕ and ⊗.

• 6. The field(F,  ⊕ , ⊗) is given as

$$ F=\left\{0,1\right\} $$

$$ \oplus \to \mathrm{Mod}-2\ \mathrm{addition} $$

$$ \otimes \to \mathrm{Mod}-2\ \mathrm{multiplication}. $$

• Using the field elements, we construct vectors, and using these vectors, we form a vector set as in

  $$ \mathbf{\mathcal{V}}=\left[0000\kern0.5em 1011\kern0.5em 0101\ 1110\ \right]. $$

• Determine whether $\mathbf{\mathcal{V}}$ is a vector space or not.

• 7. The dimension of a vector space constructed using the elements of the binary field is 4. Write all the elements of the vector space, and find three different bases of this vector space.

• 8. The dimension of a vector space constructed using the elements of the binary field is 5. Find a basis of this vector space other than the standard basis, and using the basis, determine a subspace of the vector space.

• 9. Using the elements of the prime field F₃ = {0, 1, 2}, construct a basis of a vector space whose dimension is 3, and determine all the elements of the vector space.

• 10. The basis of a vector space constructed using the binary field is given as

$$ \boldsymbol{B}=\left[00101\ 01010\ 10010\ 10101\right]. $$

• Find the elements of the vector space, and find a subspace of this vector space.

• 11. Explain the difference between linear independence and orthogonality.