{'fraction_non_alphanumeric': 0.05731300418127296, 'fraction_numerical': 0.010347594712167927, 'mean_word_length': 3.6191962543893874, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 2, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 25, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Pasting and Reversing Operations over some Vector Spaces

Pasting and Reversing operations have been used successfully over the set of integer numbers, simple permutations, rings and recently over a generalized vector product. In this paper, these operations are defined from a natural way to be applied over vector spaces. In particular we study Pasting and Reversing over vectors, matrices and we rewrite some properties for polynomials as vector space. Finally we study some properties of Reversing through linear transformations as for example an analysis of palindromic and antipalindromic vector subspaces.


Introduction
Pasting and Reversing are natural processes that the people do day after day, we paste two objects when we put them together as one object, and we reverse one object when we reflect it over a symmetry axis. We can apply these processes over words, thus, Pasting of lumber with jack is lumberjack, while Reversing of lumber is rebmul. A celebrate phrase of Albert Einstein is I prefer pi, in which Reversing of this phrase is itself and for instance this is a palindromic phrase, another palindromic phrases can be found in http://www.palindromelist.net.
Similarly to palindromic phrases, we can think in palindromic poetries, where each line can be palindromic or the hole poetry is palindromic. The following poems can be found at http://www.trauerfreuart.de/palindrome-poems.htm One mathematical theory to express these processes as operations was developed by the first author in [1,2], followed recently by [3,4,5].
In [2] is introduced the concept Pasting of positive integers to obtain their squares as well their squares roots. Five years later, in [1], are defined in a general way the concepts Pasting and Reversing to obtain genealogies of simple permutations in the right block of Sarkovskii order which contains the powers of two. Two years later, in [4], were applied Pasting and Reversing, as well palindromicity and antipalindromicity, over the ring of polynomials, differential rings and mathematical games incoming from M. Tahan's book The man who counted. Another approaches for reversed polynomials, palindromic polynomials and antipalindromic polynomials can be found in [6,11]. One year later, in [3], is applied Reversing over matrices to study a generalized vector product, in particular were studied relationships between palindromicity and antipalindromicity with such generalized vector product. Finally, in the preprint [5] were applied Pasting and Reversing over simple permutations with mixed order 4n + 2, following [1].
The aim of this paper is to study Pasting and Reversing, as well palindromicity and antipalindromicity, over vector spaces (vectors, matrices, polynomials, etc.). Some properties are analyzed for vectors, matrices and polynomials as vector spaces, in particular we prove that W a ⊂ V (set of antipalindromic vectors of V ) and W p ⊂ V (set of palindromic vectors of V ) are vector subspaces of V (V is a vector space). Moreover, V = W a ⊕ W p and in consequence dim(V ) = dim(W a ) + dim(W p ).
The reader does not need a high mathematical level to understand this paper, is enough with a basic knowledge of linear algebra and matrix theory, see for example the books given in references [7,10]. Finally, as butterfly effect, we hope that the results and approaches presented here can be used and implemented in the teaching of basic linear algebra for undergraduate level.

Pasting and Reversing over Vectors
In this section we study Pasting and Reversing over vectors in two different approaches, the first one corresponds to an analysis without linear transfomatios, using basic definitions and properties of vectors. We consider the field K and the vector space V = K n .

A first study without linear transformations.
Here we study Pasting and Reversing over vectors using the basic definitions and properties of vectors. In this way, any student beginner of linear algebra can understand the results presented. We start giving the definition of Reversing.
(3) Let assume v = (v 1 , v 2 , . . . , v n ) and w = (w 1 , w 2 , . . . , w n ), thus product between v and w is given by Now, by properties of determinants (interchanging rows and columns) we obtain Definition 1.1 lead us to the following definition.  (1) The sum of two palindromic vectors belonging to K n is a palindromic vector belonging to K n . (2) The sum of two antipalindromic vectors belonging to K n is an antipalindromic vector belonging to K n . (3) The vector product of two palindromic vectors belonging to K 3 is an antipalindromic vector belonging to K 3 . (4) The vector product of two antipalindromic vectors belonging to K 3 is the vector (0, 0, 0). (5) The vector product of one palindromic vector belonging to K 3 with one antipalindromic vector belonging to K 3 is a palindromic vector belonging to K 3 .
Proof. We prove each statement according to its item.
(1) Let v and w be palindromic vectors. We have that v + w = v + w = v + w.
In consequence, v + w is a palindromic vector. (2) Let v and w be antipalindromic vectors. We have that In consequence, v + w is an antipalindromic vector.
Remark 1.5. In [3] were studied, in a more general way, the vector product for palindromic and antipalindromic vectors. There was used a generalized vector product and were obtained some results involving the palindromicity and antipalindromicity of vectors. For completeness we present in section 3 such results with proofs in detail. Now we proceed to introduce the concept of Pasting over vectors.
Definition 1.6. Consider v ∈ K n and w ∈ K m , then v ⋄ w is given by The following properties are consequences of Definition 1.6.
. . , c m } basis of K n and K m respectively. Due to v ∈ K n and w ∈ K n , we have by Definition 1.6 that v ⋄ w ∈ K n+m , then there exists a basis B n+m = {d 1 , d 2 , . . . , d n+m } belonging to K n+m , for instance K n ⋄ K m ∼ = K n+m .
Again, in virtue of Definition 1.6, we have that thus we can conclude that Proposition 1.10. Let V = K n be a vector space. Consider W p and W a as the sets of palindromic and antipalindromic vectors of V respectively. The following statements hold.
(1) W p is a vector subspace of V , Proof. We consider each statement separately.
(1) Suppose that a, b ∈ K and v, w ∈ W p . For instance we have that v =ṽ and w =w. By Proposition 1.2 we observe that av + bw = aṽ + bw = av + bw ∈ W p , in consequence, W p is a vector subspace of V .
(2) We analyze the cases when n is even and also when n is odd.
(a) Consider V = K n and we start assuming that n = 2k. If v ∈ W p , then In this way we write the vector v as follows: The set of vectors of the previous linear combination are palindromic and linearly independent vectors, for instance they are a basis for W p and in consequence (b) Consider V = K n and now we assume that n = 2k − 1. If v ∈ W p , then Thus, we have that that is, k − 1 pairs plus the fixed component v k . This lead us to express the vector v as follows and for instance we have that The set of vectors of the previous linear combination are palindromic and linearly independent vectors, for instance they are a basis for W p and in consequence In this way, we have proven that for all n ∈ Z + , dim and for instance av +bw ∈ W a , which implies that W a is a vector subspace of V . (4) We analyse the cases when n is even as well when n is odd.
(a) Consider V = K n and we can start assuming that n = 2k. If v ∈ W a , then The set of vectors of the previous linear combination are antipalindromic and linearly independent vectors, for instance they are a basis for W a and in consequence (b) Consider V = K n and now we suppose that n = 2k − 1. If v ∈ W a , then Thus, we obtain that that is, k − 1 pairs plus the fixed component v k = 0. This lead us to express the vector v as follows: and for instance we have that The set of vectors of the previous linear combination are antipalindromic and linearly independent vectors, for instance they are a basis for W a and in consequence In this way we have proven that for all n ∈ Z + , dim W a = n 2 .
In the same way we can observe that w a = v − v 2 is an antipalindromic vector and for instance v = w p + w a , ∀v ∈ V .
1.2. Reversing as linear transformation. We denote by R the transformation Reversing, i.e., for all v ∈ K n , Rv = v. Thus, we summarize some previous results in the next proposition.
Proposition 1.11. The following statements hold (1) R is a linear transformation.
(2) The transformation matrix of R is given by (3) R is an automorphism of K n .
(4) Minimal and characteristic polynomial of R are given respectively by be a palindromic vector and let F a (v) be an antipalindromic vector. Then F p and F a (Palindromicing and Antipalindromicing transformations respectively) are isomorphisms from K n to ker(R − id) and from K n to ker(R + id) respectively. Furthermore, Proof. We proceed according to each item.
(1) For any scalar α ∈ K and any pair of vectors v, w ∈ K n we obtain being I n and I n (Reversing of the identity matrix and the identity matrix of size n × n) given by (3) Owing to R : K n → K n and v = v implies that R is left-right invertible, then R is an automorphism of K n . (4) We observe that R 2 −id = 0, being id the identity tranformation, therefore Q(λ) = λ 2 − 1 is the minimal polynomial of R, that is, Q( I n ) = 0 n ∈ M n×m (0 n is the zero matrix of size n × n). For instance, Q divides to the characteristic polynomial of R which is given by Palindromicing and Antipalindromicing transformations, denoted by F p and F a respectively, are given by Due to R and id are linear transformations defined over K n , then are linear transformations over K n . Owing to ker(F p ) = ker(F a ) = 0 ∈ K n , then F p and F a are monomorphisms. In the same way, due to Im(F p ) = ker(R − id) and Im(F a ) = ker(R + id), then F p and F a are epimorphisms. For instance, F p and F a are isomorphisms from K n to ker(R − id) and from K n to ker(R + id) respectively. Finally, we can see that any vector v ∈ K n can be expressed as the sum of a palindromic vector (obtained through F p ) with an antipalindromic vector (obtained through F a ). That is, F p (v) is a palindromic vector and F a (v) is an antipalindromic vector, furthermore, v = F p (v) + F a (v).
Remark 1.12. can see R as a particular case of a linear transformation associated to a permutation matrix. Recall that A σ is a permutation matrix, defined over a given σ ∈ S n , whether its associated linear transformation R σ is given by This means that Reversing corresponds to R σ = R, where the permutation σ and the matrix A σ are given respectively by

Pasting and Reversing over Polynomials
In [4] we studied Pasting and Reversing over polynomials from an different approach, we studied these operations focusing on the ring structure for polynomials. In this section we rewrite some properties of Pasting and Reversing over polynomials, but considering to the polynomials as a vector space. Thus, we apply the previous results for vectors, which we gave in Section 1.
Along this section we consider (K n [x], +, ·) as the vector space of the polynomials of degree less than or equal to n over the field K. This vector space is isomorphic to (K n+1 , +, ·). In this context we do not impose conditions over the polynomials just like the conditions given in [4], for example, we do not need that x ∤ P (x). The following result summarizes the properties given in Section 1 for polynomials.
, the following statements hold.
The sum of two palindromic polynomials of degree n is a palindromic polynomial of degree n. (6) The sum of two antipalindromic polynomials of degree n is an antipalindromic polynomial of degree n.
Proof. Owing to (K n [x], +, ·) ∼ = (K n+1 , +, ·) as vector spaces we apply R and ⋄ over the polynomials as vectors. The proof is done using Proposition 1.11 and properties of Pasting proven in Section 1.1.
Remark 2.2. As we can see, this section is a rewriting of Section 1 without new results for polynomials as vector space, only we suggest the proofs based on the definition and properties of R. Another interesting thing of this section is that we recover some results given in [4,11].

Pasting and Reversing over Matrices
In this section we consider the vector space M n×m (matrices of size n × m with elements belonging to K) which is isomorphic to K nm . We present here different approaches for Pasting and Reversing.

Pasting and Reversing by rows or columns.
We can see any matrix as a row vector of its column vectors as well as a column vector of its row vectors. Thus, to matrices we can introduce Pasting and Reversing by rows and columns respectively. Let us denote by A r Reversing of the row vectors v i ∈ K n of A and by A c Reversing of the column vectors c j ∈ K m of A, where 1 ≤ i ≤ m and 1 ≤ j ≤ n. For instance, Let us denote by R r A := A r and R c A := A c , where R r and R c for suitability will be called r-Reversing and c-Reversing respectively. Owing to v i = v i I m and c i = I n c j for 1 ≤ i ≤ n, 1 ≤ j ≤ m we obtain that A r = A I m and A c = I n A. Therefore R r A = A I m and R c A = I n A and for instance we can define palindromicity and antipalindromicity by rows and columns respectively. Now, we can assume A ∈ M n×m (K), B ∈ M q×m (K) and C ∈ M n×p (K) given as follows.
As we can see, in agreement with Section 1, we transformed the column vectors f j , g j and h j in the form of row vectors through the transposition of matrices (f T j , g T j and h T j are row vectors). Thus, we can define both Pasting by rows (denoted by ⋄ r ) over the matrices A and C and Pasting by columns (denoted by ⋄ c ) over the matrices A and B as follows.
From now on we paste column vectors directly without the use of trasposition of vectors. Thus, Pasting of column vectors f i and g i is In the same way, if T = M n×m (K) and S = M l×m (K), then T ⋄ S = M (n+l)×m (K). (6) Let W r p and W c p be the set of palindromic matrices by rows and columns of M n×m (K) respectively, then the sets W r p and W c p are vector subspaces of M n×m (K). (7) dim W r p = n m 2 , dim W c p = m n 2 . (8) Let W r a and W c a be the set of antipalindromic matrices by rows and columns of M n×m (K) respectively, then then W r a and W c a are vector subspaces of M n×m (K) (9) dim W r a = n m 2 , dim W c a = m n 2 (10) The sum of two palindromic matrices by rows (resp. by columns) of the same vector space is a palindromic matrix by rows (resp. by columns). (11) The sum of two antipalindromic matrices by rows (resp. by columns) of the same vector space is an antipalindromic matrix by rows (resp. by columns).
Proof. From (1) to (13) we proceed as in the proofs of Section 1 using the properties of R. (14) is consequence of the definition of Pasting by rows and columns.
Remark 3.2. There are a lot of mathematical software in where Pasting of matrices is very easy, for example, in Matlab Pasting by rows is very easy: [A, B], as well by columns [A; B], however we can build our own program using our approach given in the previous proposition, following the same structure of Pasting of polynomials as in [4]. Thus, we paste matrices by rows and columns using the item (14) in Proposition 3.1. The interested reader can proof the statements of this paper concerning to Pasting using such equations.
The following proposition summarizes some properties derived from Pasting and Reversing by rows and columns with respect to classical matrix operations. ( The product of two palindromic matrices by rows (resp. by columns) is a palindromic matrix by rows (resp. by columns). (7) The product of two antipalindromic matrices by rows (resp. by columns) is an antipalindromic matrix by rows (resp. by columns). (8) AB = 0 is a palindromic matrix by rows (resp. AB = 0 is a palindromic matrix by columns) if and only if B is a palindromic matrix by rows (resp. A is a palindromic matrix by columns). (9) AB = 0 is an antipalindromic matrix by rows (resp. AB = 0 is an antipalindromic matrix by columns) if and only if B is an antipalindromic matrix by rows (resp. A is an antipalindromic matrix by columns).
Proof. We proceed according to each item (1) We see that R c A = A I n = ( I T n A T ) T = ( I n A T ) T = (R r A T ) T and for instance R r A T = (R c A) T . In the same way, we see that R r A = I n A = (A T I T n ) T = (A T I n ) T = (R c A T ) T and for instance R c A T = (R r A) T .
(2) Assume A ∈ M n×m (K) and B ∈ M n×p (K). Let v i and w i be the row vectors of A and B respectively. Thus v i ⋄ w i , i = 1, . . . , n, are the row In the same way we can assume A ∈ M n×m (K) and B ∈ M p×m (K). Let c j and d j be the column vectors of A and B respectively. Therefore c j ⋄ d j , j = 1, . . . , m, are the column vectors of A⋄ c B, then (c j ⋄d j ) T = c T j ⋄d T j are the row vectors of (A⋄ c B) T = A T ⋄ r B T .  ). Now, it is follows by induction that we can transform I n into I n throught ⌊ n 2 ⌋ elementary operations, therefore, det( In the same way, asum- In the same way, asumming R c A = −A and R c B = −B we obtain R c (AB) = AR c B = −AB. (8) Assume A ∈ M n×m (K) and B ∈ M m×p (K), being AB = 0. By previous item we see that if A is palindromic by rows (resp. if B is palindromic by columns), then AB is palindromic by rows (resp. then AB is palindromic by columns). Now, suppose that R r (AB) = AB = 0, for instance (R r A)B = ( I n A)B = AB, which implies that R r A = A. In the same way, if we asumme R c (AB) = AB = 0, for instance (R c A)B = A( I n B) = AB = 0, which implies that R c B = B. (9) Assume A ∈ M n×m (K) and B ∈ M m×p (K), being AB = 0. By previous item we see that if A is antipalindromic by rows (resp. if B is antipalindromic by columns), then AB is antipalindromic by rows (resp. then AB is antipalindromic by columns). Now, suppose that R r (AB) = −AB = 0, for instance (R r A)B = ( I n A)B = −AB, which implies that R r A = −A.
In the same way, if we asumme R c (AB) = −AB = 0, for instance Now, in a natural way, we can introduce the sets W pp := W r p ∩ W c p , W pa := W r p ∩ W c a , W ap := W r a ∩ W c p and W aa := W r a ∩ W c a . The sets W pp and W aa correspond to the set of double palindromic matrices and the set of double antipalindromic matrices respectively.
Proposition 3.4. The following statements hold.
(1) W pp , W pa , W ap and W aa are vector subspaces of M n×m (K).
Proof. The intersection of vector subspaces is a vector subspace. The rest is obtained through elementary properties of vector subspaces and by the nature of W pp , W pa , W ap and W aa .
Finally, according to Remark 1.5, we present the results concerning to the relationship between Reversing and the generalized vector product of n − 1 vectors of K n , which are given in [3, §3].
Let M 1 = (m 11 , m 12 , . . . , m 1n ) , . . . , M n−1 = m (n−1),1 , a (n−1),2 , . . . , m (n−1),n , be n − 1 vectors belonging to K n . It is known that the generalized vector product of these vectors is given by being e k the k-th element of the canonical basis for K n and M (k) is the square matrix obtained after the deleting of the k-th column of the matrix M = (m ij ) ∈ M (n−1)×n (K), for more information see [3,10]. Therefore, the matrix M (k) is a square matrix of size (n − 1) × (n − 1) and is given by Proposition 3.5. Consider the matrix M = (m ij ) ∈ M (n−1)×n (K), then R r (M (k) ) = M (n−k+1) I n−1 , for 1 ≤ k ≤ n.
On the other hand, for instance R r (M (k) ) = M (n−k+1) I n−1 , for 1 ≤ k ≤ n. Thus we conclude the proof. This means that the generalized vector product of (n − 1) palindromic vectors belonging to K n is interesting whenever 1 ≤ n ≤ 3. The same result is obtained when we assume M as an antipalindromic matrix by rows, so we recover the previous results given in Section 1 with respect to the vector product. Moreover, some rows of M can be palindromic vectors, while the rest can be antipalindromic vectors, in this way, we can obtain similar results.
We come back to express RA and A ⋄ B in term of matrices instead of vectors, i.e., We say that any matrix P , with the conditions established above, is a palindromic matrix whether P = RP = P . In the same way, we say that any matrix A, with the conditions established above, is an antipalindromic matrix whether A = RA = −A.
Note that R(I n ) = I n where I n is written in the matrix form, but it should keep in mind that R(I n ) = I n I n 2 , where I n is written as vector. Thus, we arrive to the following elementary result.
Proof. It is followed by definition of Reversing in matrices seen as vectors.
The following proposition summarizes the previous results for matrices as vectors.
Proposition 3.9. Consider A ∈ M n×m (K), B ∈ M p×q (K) and C ∈ M r×s (K) satisfying the conditions established above, the following statements hold.
(1) (6) Let W p be the set of palindromic matrices of M n×m (K), then W p is a vector subspace of M n×m (K). (7) dim W p = nm 2 (8) Let W a be the set of antipalindromic matrices of M n×m (K), then W a is a vector subspace of M n×m (K). (9) dim W a = nm 2 (10) The sum of two palindromic matrices of the same vector space is a palindromic matrix. (11) The sum of two antipalindromic matrices of the same vector space is an antipalindromic matrix.
Proof. Proceed as in the proofs of Section 1 using Lemma 3.8.
The following result shows the relationship of Reversing with matrices classical operations.
Proposition 3.10. The following statements hold.
( (2) Assume A ∈ M n×m (K) and B ∈ M m×r (K). Thus, AB ∈ M n×r (K), which implies that C = A B and then R(AB) = R(A)R(B). At this point, we have considered Pasting over an special case of matrices. As we can see, it can be confused when both matrices have the same size, how can we paste them? Another natural question is: how can we paste to matrices whenever n = p and m = q? To avoid this difficulty we introduce Pasting by blocks, which will be denoted by ⋄ b . Consider matrices A ∈ M n×m (K) and B ∈ M r×s (K), Pasting by blocks of A with B is given by It is well known that Pasting by blocks corresponds to a particular case of block matrices, also called partitioned matrices, see [8,9]. The following result, although is known from block matrices point of view, is consequence of Proposition 3.9 and Proposition 3.10 considering R as above and ⋄ b instead of ⋄.
Proposition 3.11. Consider the matrices A ∈ M n×m (K), B ∈ M p×q (K) and C ∈ M r×s (K), the following statements hold.
(  Proof. According to each item: (1) follow directly from the definition of Pasting by blocks, Reversing and Pasting of vectors.
(2) is due to A ⋄ b B = A ⊕ B.

Final Remarks
This paper is an invitation to undergraduate students and teachers of linear algebra to explore Pasting and Reversing in advanced topics of linear algebra, as well in other branches of mathematics and physics. Here we solved one question proposed in [4], which relates Pasting and Reversing with vector spaces and matrix theory. However, there are a lot of questions about the applications of Pasting and Reversing, as well generalizations of these operations. The interested reader can try to solve the following natural questions.
• What properties for Pasting and Reversing hold when we use A σ as the companion matrix of the linear transformation R σ , where σ ∈ S n and σ 2 = e? In particular, what happens when R n σ = e, being R n−1 σ = e? • How can we apply Pasting and Reversing in mathematical physics and dynamical systems? It is easy to see that Pauli spin matrices can be written in term of Reversing and the symplectic matrix can be written in term of Pasting. Furthermore, for reversible systems this approch has been applied succesfully, see [11].