Algebraic Structures

Definitions and Examples

An algebraic structure \(\,\) is by definition a collection of a finite number of sets and a finite number of mappings of Cartesian products of these sets into these sets.

To phrase that in a more straightforward way, an algebraic structure consists of one or several sets and some operations (a set the operations are defined upon is named an underlying set of the given structure). The definition of a particular structure specifies the underlying set(s) together with the operations and gives a list of axioms to be fulfilled by these operations.

The most important algebraic structures are: \(\,\) group, \(\,\) ring, \(\,\) field, \(\,\) vector space, \(\,\) algebra.

A \(\ \) Group \(\ \,\mathbb{G}\ \,\) consists of one set \(\ G\ \) and \(\,\) one internal operation \(\ \bot:\)

\(\mathbb{G}\;=\;(G,\,\bot\,)\,,\ \ \) where \(\ \ \bot:\ \ G\times G \to G.\)

A group operation \(\ \bot\ \) satisfies the following conditions: 1

1. \(\,\) is associative:

   \[(g_1\,\bot\;g_2)\ \bot\ g_3\ \ =\ \ g_1\ \bot\ (g_2\;\bot\;g_3)\qquad \forall\ \ g_1,\,g_2,\,g_3\in G\,;\]

2. \(\,\) there exists a neutral element \(\ e\in G\ \,\) such that

   \[e\;\bot\;g\ \,=\ \,g\;\bot\;e\ \,=\ \,g \qquad \forall\ g\in G\,;\]

3. \(\,\) each element \(\ g\in G\ \) has its inverse \(\ g^{-1}\in\,G\ \) such that

   \[g\;\bot\;g^{-1}\ \,=\ \,g^{-1}\;\bot\;g\ \,=\ \,e\,.\]

If additionally the operation \(\ \,\bot\,\ \) is commutative:

\[g_1\,\bot\;g_2 \ \,=\ \,g_2\;\bot\;g_1\qquad\forall\ \ g_1,\,g_2\,\in\,G\,,\]

the group is called \(\,\) abelian. In an abelian group the operation is often symbolized by ‘+’ \(\\\) and the term ‘inverse’ is replaced by ‘opposite’: \(\ \ g^{-1}\rightarrow (-g).\)

Examples of abelian groups.

• Additive groups of numbers \(\,(X,\,+),\ \) where \(\ X = Z,\,Q,\,R\ \) or \(\ C\).

• The additive group of integers modulo \(\,n\in N:\ \ (Z_n,\,+_n)\,;\)

  here \(\,Z_n\,:=\{0,1,\ldots,n-1\}\ \) and \(\,+_n\,\) symbolizes the addition modulo \(n\).

• Multiplicative groups of nonzero numbers: \(\,(K\!\smallsetminus\!\{0\},\,\cdot\,),\ \) where \(\ K=Q,\,R,\,C\).

• The additive group of geometric vectors in a plane or in the space: \(\ (V,\,+\,)\).

Examples of non-abelian groups.

• A group of permutations of an \(\,n\)-element set \(\,(S_n,\,\circ\,)\)

  (the group operation \(\,\circ\,\) is the composition of permutations as functions);
• The group of real nonsingular square matrices of size \(\,n\)

  (with matrix multiplication as the group operation).
• A symmetry group of a figure in a plane or of a solid figure in the space

  (with composing the symmetry operations).

A \(\ \) Ring \(\ \,\mathbb{P}\ \,\) consists of one set \(\,P\ \) and \(\,\) two internal operations:

\(\ \ \mathbb{P}\;=\;(P,\,+\,,\,\cdot\,)\,,\ \ \) where \(\ \ \,+\,,\ \cdot\ :\ \,P\times P \to P\,\)

(the operations shall be henceforth called addition and multiplication, respectively).

Ring operations satisfy the following conditions:

1. \(\ \) the structure \(\ (P,\,+)\ \ \) is an abelian group

   \(\ \) (the additive group of the ring; its neutral element is the zero of the ring);

   \(\ \)

2. \(\ \) multiplication is associative:

   \[(a_1\cdot a_2)\cdot a_3 \ =\ a_1\cdot (a_2\cdot a_3)\qquad \forall\ \ a_1,\,a_2,\,a_3\,\in\,P\,;\]

3. \(\ \) multiplication is distributive over addition:

   \[ \begin{align}\begin{aligned}a\cdot(b_1+b_2)\ =\ (a\cdot b_1)\ +\ (a\cdot b_2)\qquad \forall\ \ a,\,b_1,\,b_2\,\in\, P\,,\\(a_1+a_2)\cdot b\ =\ (a_1\cdot b)\ +\ (a_2\cdot b)\qquad \forall\ \ a_1,\,a_2,\,b\,\in\, P\,.\end{aligned}\end{align} \]

When, additionally, multiplication is commutative:

\[a_1\cdot\,a_2 \ =\ a_2\cdot\,a_1\qquad \forall\ \ a_1,\,a_2\,\in\,P,\]

the ring itself is said to be commutative.

Furthermore, if there exists in \(\,P\,\) a multiplicative neutral element \(\\\) (named the identity of the ring and denoted by 1):

\[1\cdot a\ =\ a\cdot 1\ =\ a\qquad\forall\ a\in P\,,\]

then we are dealing with a ring with identity.

Examples of commutative rings with identity.

• The integer ring \(\ \mathbb{Z} \,=\, (Z,\ +\,,\ \cdot\,)\ \) with the usual addition and multiplication of numbers.

• The ring of integers modulo \(\,n>1:\,\) \(\ \mathbb{Z}_n = (Z_n,\ +_n\,,\ \cdot_n\,)\,,\ \) where \(\ Z_n\,=\,\{0,1,\ldots,n-1\}\ \)

  and the operations \(\ \,+_n\ \ \) and \(\ \ \cdot_n\ \,\) are the addition and multiplication modulo \(\,n\).
• The structure \(\,(X,\ \oplus,\ \odot\,)\,,\ \) where \(\,X\,=\,Z,\,Q,\,R\ \) or \(\ C,\ \) and the operations are

  defined as follows: \(\quad a\oplus b\,:\,=\, a+b+1,\quad a\odot b\,:\,=\,a+b+ab,\quad a,b\in X\).

  Interestingly enough, the zero (additive neutral element) is here the number \(\,-1,\,\)

  whereas the identity (multiplicative neutral element) is the number \(\,0\).
• The set \(\,\mathcal{F}_{[\,0,\,1\,]}\,\) of all functions mapping the interval \(\ [\,0,\,1\,]\ \) into \(\ R\),

  in conjunction with the operations of adding and multiplying functions,

  forms a ring in which the zero is the function \(\ \,\theta(x)=0,\ \ x\in [\,0,\,1\,]\,,\)

  and the identity is the function \(\ e(x)=1,\ \ x\in [\,0,\,1\,]\,.\)

An example of noncommutative ring is given by the set of square matrices of size \(\,n>1\,\) over a ring with identity, considered together with the operations of matrix addition and multiplication. \(\\\)

A \(\ \) Field \(\ \mathbb{K},\ \) just as a ring, is composed of one set \(\,K\ \) and two internal operations, \(\\\) \(\ \,+\ \,\) and \(\ \,\cdot\ \,,\ \ \) named respectively addition and multiplication: \(\ \ \mathbb{K}\;=\;(K,\,+\,,\,\cdot\,)\,.\)

By definition, the field operations are subject to the following conditions:

1. \(\ (K,\,+)\ \ \) is an abelian group with the neutral element \(\, 0 \,\) (zero of the field);

2. \(\ (K\!\smallsetminus\!\{0\},\ \cdot\;)\ \ \) is an abelian group with the neutral element \(\, 1 \,\) (identity of the field);

3. \(\ \) multiplication is distributive over addition:

   \[\alpha\cdot(\beta_1+\beta_2)\ =\ (\alpha\cdot \beta_1)\ +\ (\alpha\cdot \beta_2)\qquad \forall\ \ \alpha,\,\beta_1,\,\beta_2\,\in\, K\,.\]

\((K,\,+)\ \) and \(\ K\!\smallsetminus\!\{0\},\ \cdot\;)\ \) are the additive group and multiplicative group of the field, resp. \(\\\) It’s easy to note that every field is a ring (though the reverse is not true). More precisely: \(\\\) a field is a commutative ring with identity, in which each nonzero element has an inverse.

Examples of fields.

• Field of rational numbers: \(\ \mathbb{Q}\,=\,(Q,\,+\,,\;\cdot\;)\;;\ \)

  field of real numbers: \(\ \mathbb{R}\,=\,(R,\,+\,,\;\cdot\;)\;;\ \)

  field of complex numbers: \(\ \mathbb{C}\,=\,(C,\,+\,,\;\cdot\;)\,.\)
• The set \(\,Q(\sqrt{2})\,:\,=\,\{\,a+b\sqrt{2}\,:\ a,b\in Q\,\}\)

  forms a field with the usual operations on real numbers.
• The structure \(\,(X,\ \oplus,\ \odot\,)\,,\ \) where \(\,X\,=\,Q,\,R\ \) or \(\,C,\ \) and the operations are

  defined as follows: \(\quad a\oplus b\,:\,=\, a+b+1,\quad a\odot b\,:\,=\,a+b+ab,\quad a,b\in X\),

  is a field \(\ \) (here the zero is the number \(\,-1\,\) and the identity is the number \(\,0\,\)).
• The ring \(\ \mathbb{Z}_n\ \) of integers modulo \(\,n\ \) is \(\,\) a (finite) field \(\,\) if and only if \(\,n\,\) is a prime.

Basic properties of fields are discussed in Appendix A1.

Note. \(\;\)

In a less rigorous (yet widely used) language, an algebraic structure is often identified with its underlying set. \(\\\) So one says \(\ \) “\(\ \sigma\,\) is an element of the permutation group \(\,S_n\,\)” \(\ \) \(\\\) or \(\,\) “the set \(\,2Z\,\) of even integers is a commutative ring without identity”.

A \(\ \) Vector Space \(\,\) over a field \(\,K\ \ \) is the structure \(\ \ \mathbb{V}\,= \,(\,V,\,\oplus\,;\ \,K,\,+\,,\,\cdot\ \,;\ \;\boxdot\,)\,,\ \)

where \(\quad\oplus:\ V\times V\rightarrow V\,; \quad +\ ,\ \cdot\,:\ K\times K\rightarrow K\,; \quad\boxdot\,:\ K\times V\rightarrow V\,.\)

By definition, the operations satisfy the following requirements:

1. \(\ (V,\,\oplus\,)\ \ \) is an abelian group

   \(\ \) (the additive group of the vector space; its neutral element is the zero vector \(\,\theta\));

2. \(\ (K,\,+\,,\,\cdot\,)\ \ \) is a field;

3. \(\ (\alpha + \beta)\,\boxdot\,v \ \,=\ \, (\alpha\,\boxdot\,v)\,\oplus\,(\beta\,\boxdot\,v),\)

   \(\ \ \alpha\,\boxdot\,(v\,\oplus\,w) \ \,=\ \, (\alpha\,\boxdot\,v)\ \oplus\ (\alpha\,\boxdot\,w);\)

4. \(\ \ \alpha\,\boxdot\,(\beta\,\boxdot\,v) \ \,= \ \,(\alpha\cdot\beta)\,\boxdot\,v;\)

5. \(\ \ 1\ \boxdot\ v \ \,=\ \,v.\)

In the above expressions \(\ \alpha\ \) and \(\ \beta\ \) are arbitrary elements of the field \(\,K\ \) \(\\\) (\(\,1\,\) is the identity of that field), \(\ \) whereas \(\ v\ \) and \(\ w\ \) are arbitrary elements of the set \(\,V.\)

Elements of the set \(\,K\ \) are called scalars, \(\,\) while those from the set \(\,V\) are vectors.

As mentioned in the Note above, in a practical mathematical language the term “vector space” would refer to the set \(\,V\,\) rather than to the algebraic structure \(\ \mathbb{V}\,\) as a whole. Assuming this terminology, we may define a vector space in a less formal way as a collection \(\,V\,\) of vectors, which may be added (to form an abelian group) and multiplied by scalars from a field, the multiplication being distributive over both addition of scalars and addition of vectors, being also compatible with the field multiplication and satisfying the trivial condition \(\,1\cdot v=v,\ v\in V\).

A vector space over a field \(\,K\,\) will be denoted shortly by \(\,V(K).\) \(\\\) Usually, the field \(\,K\,\) is either the field of real numbers \(\,R\,\) or the field of complex numbers \(\,C.\) \(\\\) Accordingly, we talk about a real vector space \(\,V(R)\,\) \(\,\) or \(\,\) of a complex one \(\,V(C).\)

Examples of vector spaces.

• Set of real numbers \(\,R\,\) yields the vector space \(\,R(Q)\,\) over the field of rational numbers \(\,Q\,,\ \) as well as the vector space \(\,R(R)\,\) over the field of real numbers \(\,R\,.\)

  On an analogous basis, the set of complex numbers \(\,C\,\) forms the real space \(\,C(R)\) and the complex space \(\,C(C)\,.\)

• Set of geometric vectors in a plane or in the space is a real vector space with respect to vector addition and multiplication by real numbers.

• Set \(\,K^n\,\) of \(\,n\)-tuples, represented by column vectors of size \(\,n,\ \) is a vector space under the vector addition and scalar multiplication.

• Set \(\,M_{m\times n}(K)\,\) of rectangular matrices over a field \(\,K\ \) with \(\,m\,\) rows and \(\,n\,\) columns

  is a vector space over the field \(\,K\ \) under the matrix addition and scalar multiplication.

Assuming the usual order of operations we shall apply henceforth the simplified notation:

\[(\alpha\,\boxdot\,v)\ \oplus\ (\beta\,\boxdot\,w)\quad\rightarrow\quad \alpha\cdot v\,+\,\beta\cdot w\quad\rightarrow\quad \alpha\,v\,+\,\beta\,w\,.\]

Basic properties of vector spaces are described in Appendix A2. \(\\\) A vector space over the field \(\,R\ \) or \(\,C\ \) is a fundamental object of linear algebra. However, the computer algebraic system Sage is based on a more general concept of a module over a ring.

A \(\ \) left Module \(\,\) over a ring \(\,P\ \ \) is the structure \(\ \ \mathbb{M}\,=\, (\,M,\,\oplus\,;\ \,P,\,+\,,\,\cdot\ \,;\ \;\boxdot\,)\,,\ \)

where \(\quad\oplus:\ M\times M\rightarrow M\,; \quad +\ ,\ \cdot\,:\ P\times P\rightarrow P\,; \quad\boxdot\,:\ P\times M\rightarrow M\,.\)

The operations are subject to the following axioms:

1. \(\ (M,\,\oplus\,)\ \ \) is an abelian group (called the additive group of the module);

2. \(\ (P,\,+\,,\,\cdot\,)\ \ \) is a ring with identity;

3. \(\ (a + b)\,\boxdot\,m \ \,=\ \, (a\,\boxdot\,m)\,\oplus\,(b\,\boxdot\,m)\,,\)

   \(\ \ a\,\boxdot\,(m_1\,\oplus\,m_2) \ \,=\ \, (a\,\boxdot\,m_1)\ \oplus\ (a\,\boxdot\,m_2)\,;\)

4. \(\ \ a\,\boxdot\,(b\,\boxdot\,m) \ \,=\ \,(a\cdot b)\,\boxdot\,m\,;\)

5. \(\ \ 1\ \boxdot\ m \ \,=\ \,m\,;\qquad\quad \forall\ \ a,b\in P\,,\quad\forall\ \ m,m_1,m_2\in M.\)

A left module \(\,M\ \) over a ring \(\,P\ \ \) (in short: a left \(\,P\)-module) is therefore an abelian group, whose elements can be multiplied by scalars from \(\,P\,,\ \) the distributivity and compatibility conditions \(\,\) 3. \(-\) 5. \(\,\) being fulfilled.

Definition of a right \(\,P\)-module differs from the above by the fourth postulate:

4. \(\ \ a\,\boxdot\,(b\,\boxdot\,m) \ \,=\ \, (b\cdot a)\,\boxdot\,m\,; \qquad\forall\ \ a,b\in P\,,\quad\forall\ m\in M.\)

Then a modified notation would be more natural: \(\quad\boxdot\,:\ M\times P\rightarrow M\,,\)

3. \(\ m\,\boxdot\,(a + b)\ \,=\ \, (m\,\boxdot\,a)\,\oplus\,(m\,\boxdot\,b)\,,\)

   \(\ \ (m_1\,\oplus\,m_2)\,\boxdot\,a \ \,=\ \, (m_1\,\boxdot\,a)\ \oplus\ (m_2\,\boxdot\,a)\,;\)

4. \(\ \ (m\,\boxdot\,a)\,\boxdot\,b \ \,=\ \,m\,\boxdot\,(a\cdot b)\,;\)

5. \(\ \ m\ \boxdot\ 1 \ \,=\ \,m\,;\qquad\quad \forall\ \ a,b\in P\,,\quad\forall\ \ m,m_1,m_2\in M.\)

When a ring \(\,P\,\) is commutative, the left and right \(\,P\)-modules are identical, \(\\\) and when \(\,P\,\) is a field, the corresponding \(\,P\)-module becomes a vector space.

Examples of modules.

• A ring \(\,P\ \) is a (both left and right) module over itself.

• Given a ring \(\,P,\ \) consider the set \(\,P^{\,n}\ \) of \(\,n\)-tuples written vertically as columns. The operation of addition of tuples and that of multiplicating them by scalars from \(\,P\ \) being defined in the natural way, we obtain a left as well as a right module over \(\,P.\) \(\\\) An important example is the module \(\,Z^n\,,\ \) composed of \(\,n\)-element columns of integers. When \(\,P\ \) is a field, \(\,P=K,\ \) we get the vector space \(\,K^n.\)

• On the same lines, the set \(\,M_{m\times n}(P)\ \) of rectangular matrices over a ring \(\,P\ \) is a (left as well as right) \(\,P\)-module. In particular we get the module \(\,M_{m\times n}(Z)\ \) of matrices with integer entries.

• The set \(\,M_n(P)\ \) of square matrices of size \(\,n\ \) over a ring \(\,P\ \) is a noncommutative ring with identity with respect to matrix addition and multiplication. The left-multiplication of columns from \(\,P^{\,n}\ \) by matrices from \(\,M_n(P)\ \) is an internal operation in \(\,P^{\,n}\ \) satisfying the conditions \(\,\) 3. - 5. \(\,\) in the definition of a module. Therefore \(\,P^{\,n}\ \) is a left (and only left) module over the ring \(\,M_n(P).\)

• Every abelian group \(\,G\ \) is a module over the ring of integers \(\,Z.\ \)

An \(\ \) Algebra \(\,\) over a field \(\,K\ \ \) is the structure \(\ \ \mathbb{A}\,=\,(\,A,\,\oplus,\,\odot\,;\ \,K,\,+\ ,\, \cdot\ \,;\ \boxdot\,)\,,\ \)

where \(\quad\oplus,\,\odot:\ A\times A\rightarrow A\,; \quad +\ ,\ \cdot\,:\ K\times K\rightarrow K\,; \quad\boxdot\,:\ K\times A\rightarrow A\,.\)

The operations are bound by the following conditions:

1. \(\ (\,A,\ \oplus,\ \odot\,)\ \ \) is a ring;

2. \(\ (\,K,\ +\,,\ \cdot\ )\ \ \) is a field;

3. \(\ (\,A,\,\oplus\,;\ \,K,\,+\,,\,\cdot\ \,;\ \boxdot\,)\ \) is a vector space (over the field \(\,K\));

4. \(\ \ (\lambda\boxdot x)\,\odot\,y \ \,= \ \,x\,\odot\,(\lambda\boxdot y) \ \,= \ \,\lambda\,\boxdot\,(x\odot y)\,,\quad \forall\ \lambda\in K,\quad \forall\ \ x,y\in A\,.\)

Thus an algebra over a field \(\,K\ \) is a vector space over that field, wherein additionally vectors can be multiplied to yield a vector result. The vector multiplication is associative and distributive over the vector addition as well as compatible with the scalar multiplication of vectors in the sense of the condition 4. above.

An algebra may be characterized both as a ring or as a vector space. Specifically,

• an algebra is commutative when the vector multiplication is commutative;

• an algebra with identity contains a multiplicative neutral element;

• the basis and the dimension of an algebra refer to the respective features

  of the vector space of the algebra.

Examples of algebras.

• An arbitrary field \(\,K\,\) is a 1-dimensional commutative algebra with identity over itself.

• The set \(\,M_n(K)\,\) of square matrices of size \(\,n\,\) over a field \(\,K\,\) is a \(\,n^2\)-dimensional noncommutative algebra with identity over \(\,K\,\) under matrix addition, multiplication and scalar multiplication.

• The set \(\ \text{End}(V)\ \) of linear operators on an \(\,n\)-dimensional vector space \(\,V(K)\,\) is a \(\,n^2\)-dimensional noncommutative algebra with identity over \(\,K\,\) under operator addition, composition and scalar multiplication.

Substructures

Suppose that the structure \(\,\mathbb{G} = (G,\,\,\bot\,)\,\) is a group. It may happen that \(\\\) a subset \(\ H\,\) of the set \(\ G\,\) also forms a group under the (properly restricted) operation \(\,\bot\,\,.\ \) \(\\\) Then we say that \(\,H\,\) is a \(\,\) subgroup \(\,\) of the group \(\,G\,\ \) and write \(\ \,H\,<\,G\,.\)

Examples of subgroups.

• The set \(\,2Z\,\) of even integers is a subgroup of the additive group \(\,Z\,\) of all integers.

• The two-element set \(\ \{-1,\,1\}\ \) is a subgroup of the multiplicative group \(\\\) of nonzero real numbers.

• The set of four rotations, \(\ O_0,\,O_1,\,O_2,\,O_3\,,\ \) is a subgroup of the eight-element \(\\\) symmetry group \(\,D_4\,\) of a square in a plane.

Analogously are defined \(\,\) subrings, \(\,\) subfields, \(\,\) vector subspaces \(\,\) and \(\,\) subalgebras.

A subset of the structure’s underlying set can form a substructure only if the operations of the structure do not move elements out from this subset, that is when the subset is closed under these operations. Also, all axioms concerning the operations should be satisfied in the subset.

Fortunately, there are practical compact criteria allowing to omit a detailed verification whether each axiom is fulfilled in a given subset. For example, for groups one may easily prove (see Appendix A3) the following

Criterion for a subgroup. \(\\\)

Let \(\ \,\mathbb{G}\;=\;(G,\;\bot\,)\ \,\) be a group, \(\ \,\emptyset\neq H\,\subset G\,.\ \) Then \(\ H<G\ \) if, and only if,

\[a,b\,\in\, H\quad \Rightarrow\quad \left(\ a\;\bot\;b\ \in\ H\ \ \land\ \ a^{-1}\,\in\,H \ \right)\,.\]

(a subset \(\,H\,\) of the group \(\,G\,\) is a subgroup iff it is closed under products and inverses).

By referencing to the definition and basic properties of vector space, one may prove (see Appendix A3) the necessary and sufficient condition for a subset to be a subspace of a space.

Criterion for a vector subspace. \(\\\)

Let \(\ \,\emptyset\neq W \subset V\,,\ \) where \(\,V\,\) is a vector space over a field \(\,K.\) \(\\\) Then \(\ W < V\ \) if and only if \(\,\) for all \(\ \alpha \in K\,,\ w_1,w_2 \in V\) :

(1)\[w_1,w_2\,\in\,W \quad\Rightarrow\quad \left(\ w_1+w_2\,\in\,W\ \ \land\ \ \alpha\,w_1\,\in\,W \ \right)\,,\]

that is, \(\,\) equivalently, \(\,\) if and only if, \(\,\) for all \(\ \alpha_1,\alpha_2 \in K\,,\ w_1,w_2 \in V\) :

(2)\[w_1,w_2\,\in\,W \quad\Rightarrow\quad \alpha_1\,w_1 + \alpha_2\,w_2\,\in\,W\]

(a subset \(\,W\,\) of a vector space \(\,V(K)\,\) is a vector subspace iff \(\,W\,\) is closed under vector addition and scalar multiplication, \(\,\) that is \(\,\) iff for any two vectors from \(\,W,\ \) every linear combination of them also belongs to \(\,W\)).

Every vector space \(\,V\,\) has two improper subspaces, the whole space \(\,V\,\) and the one-element set \(\,\{\theta\}\,,\ \) where \(\,\theta\,\) is the zero vector. Other subspaces are proper.

Examples of subspaces.

• The set \(\,Q\,\) of rational numbers is a vector space \(\,Q(Q)\,\) over itself under the usual operations on numbers.

  It is a subspace of the rational space of real numbers \(\,R(Q)\,:\ \ Q(Q)<R(Q).\)
• Let \(\,V\,\) denote the set of geometric vectors in the space,

  \(\,V_x,\,V_y,\,V_z\ \,-\ \,\) subsets of vectors lying along the axes \(\,Ox,\,Oy,\,Oz\,,\) respectively,

  \(\,V_{xy},\,V_{yz},\,V_{xz}\ \,-\ \,\) subsets of vectors lying in the planes \(\,Oxy,\,Oyz,\,Oxz\,,\ \) resp.

  These subsets are subspaces of the space \(\,V:\ \ \) \(\,V_x,\,V_y,\,V_z,\,V_{xy},\,V_{yz},\,V_{xz}\,< \,V.\)

  Moreover, we note that: \(\quad V_x,\,V_y\,<\,V_{xy}\,, \quad V_y,\,V_z\,<\,V_{yz}\,,\quad V_x,\,V_z\,<\,V_{xz}\,.\)

• Consider the space \(\,K^n\ \) of column vectors of size \(\,n\,\) over a field \(\,K\,:\)

  \[\begin{split}K^n\ \ = \ \ \,\left\{\quad \left[\begin{array}{c} x_1 \\ \ldots \\ x_p \\ x_{p+1} \\ \ldots \\ x_n \end{array}\right] \ :\quad x_i\in K\,,\ \ i = 1,2,\ldots,n.\;\right\}\,,\end{split}\]

  and the subset \(\ W_p\ = \ \{\ \boldsymbol{x}\in K^n\,:\ \ x_{p+1}=\ldots = x_n = 0\,\}\,,\ \) where \(\ 1 \leq p < n\,:\)

  \[\begin{split}W_p\ \ :\,= \ \ \,\left\{\quad \left[\begin{array}{c} x_1 \\ \ldots \\ x_p \\ 0 \\ \ldots \\ 0 \end{array}\right] \ :\quad x_i\in K\,,\ \ i = 1,2,\ldots,p.\;\right\}\,.\end{split}\]

  Using the conditions (1) or (2) we easily conclude that

  \[W_p\,<\,K^n\,.\]

• The set \(\,M_n(K)\,\) of square matrices of size \(\,n\,\) over a field \(\,K\,\) is a vector space over \(\,K\,\) under matrix addition and scalar multiplication:

  \[\begin{split}M_n(K)\ \ =\ \ \left\{\quad\left[\ \begin{array}{cccc} a_{11} & a_{12} & \ldots & a_{1n} \\ a_{21} & a_{22} & \ldots & a_{2n} \\ \ldots & \ldots & \ldots & \ldots \\ a_{n1} & a_{n2} & \ldots & a_{nn} \end{array}\ \right] \ :\quad a_{ij}\in K,\ \ i,j\,=\,1,2,\ldots,n.\;\right\}\,.\end{split}\]

  The conditions (1) or (2) being satisfied, the subset composed of all diagonal matrices

  (3)\[\begin{split}D_n(K)\ \ :\,=\ \ \left\{\quad\left[\ \begin{array}{cccc} a_{11} & 0 & \ldots & 0 \\ 0 & a_{22} & \ldots & 0 \\ \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & \ldots & a_{nn} \end{array}\ \right]\ :\quad a_{ii}\in K,\ \ i\,=\,1,2,\ldots,n.\;\right\}\,.\end{split}\]

  is a subspace: \(\quad D_n(K)<M_n(K)\,.\)

The condition for a subalgebra is an extension of that for subspace.

Criterion for a subalgebra. \(\\\)

A subset \(\ B\ \) of the algebra \(\ A\ \) over a field \(\ K\ \) is a subalgebra if and only if \(\,\) it is closed under vector addition and vector multiplication as well as under scalar multiplication, that is \(\,\) iff for arbitrary \(\ x_1,x_2\in A\ \) and \(\ \lambda\in K:\)

\[x_1,x_2\,\in\,B \quad\Rightarrow\quad \left(\ x_1+x_2\,\in\,B\ \ \land\ \ x_1\,x_2\,\in B \,\ \ \land\ \ \lambda\,x_1\,\in\,B\ \right)\,.\]

On that basis the set of diagonal matrices in Eq. (3) is a subalgebra: \(\ \,D_n(K)<M_n(K)\,.\)

1

a denotement \(\ \forall\ x\in X\ \) means \(\,\) “for all \(\,x\in X\)”.