2.1 The student and professional life of Hermann Grassmann (1809 to 1843)

Hermann Günther Grassmann was born in 1809 in Stettin, at the time part of Prussia. He was a theologian, self-taught mathematician, high school teacher, and the third of twelve children of Justus Günther Grassmann and Johanne Luise Friederike Medenwald. His father, besides being a theologian, and a pietistic⁷ and pastor, was also a mathematics teacher (^{DIEUDONNÉ, 1979}).

Grassmann lived in a time when Germany did not exist as a unified nation, but was composed of 39 independent states. The most prominent and influential states within this confederation were the Austrian Empire and the Kingdom of Prussia. It is worth noting that, due to political and border changes resulting from wars and conflicts in the 20th century, Stettin is currently part of Poland. (^{ASSIS, 2018}).

During this period, Enlightenment ideals were spreading throughout the world, influencing significant social and political changes. The Enlightenment thinkers advocated values such as democracy, economic liberalism, and rationalism, which were against to monarchical absolutism and the influence of the Church over the population. The Industrial Revolution also occurred during this time, transforming society and the economy, as well as human activity.

The political and intellectual landscape of that moment likely impacted Grassmann's ideas and may have influenced his view of Mathematics and his contributions to the field, as stated by Kopnin (¹⁹⁷⁸, p. 53), “[...ϡThe laws of the objective world become laws of thought as well, and all laws of thought are represented in the objective world." With his innovative work in Algebra and Geometry, Grassmann left a mark of rigor on the development of Mathematics, contributing to the understanding and advancement of the field in a period of profound historical and intellectual changes.

Despite belonging to a family with a commendable level of education, Hermann did not stood out in the early years of secondary school, which today corresponds to elementary education. In his time, young individuals often demonstrated their aptitudes early on. Due to having attended reputable schools and received a good home education, his father believed that he could not possibly achieve any academic significance. However, gradually he began to make progress in school, eventually ranking second in the final exam of what nowadays known as high school. (^{DIEUDONNÉ, 1979}).

Determined to study theology, driven by his desire to become a Lutheran minister, in 1827 he enrolled at the University of Berlin, the oldest institution in the city and one of the most privileged in Germany and Europe. There is no information indicating that Grassmann had any academic background in Mathematics; however, it was mathematics that led him back to Stettin after completing his studies in Berlin in 1830, where he became a teacher at secondary schools and embarked on independent mathematical research (^{DIEUDONNÉ, 1979}).

Upon returning to Berlin in 1831 to take examinations for an academic career, he did not perform well, and he was only permitted to teach Mathematics at lower levels. In 1832, he was appointed as an assistant professor at the Gymnasium of Stettin, and during this period, he made his first mathematical discoveries, which would later be developed and published.

Due to the nature of his professional license, in 1835, Grassmann was appointed to teach Mathematics, Physics, German, Latin, and Religious Studies at the Otto Schule in Stettin, a primary school. Then, in 1839, he passed his theology exams, which allowed him to teach at the secondary level.

It is important here to put some emphasis on what were the duties of a Gymnasium professor in Germany at that time, and in fact all through Grassmann's lifetime: he was supposed to lecture on all subjects, from religion to biology through latin, mathematics, physics and chemistry, and at all levels, at a rate of 18 to 30 hours a week (^{DIEUDONNÉ, 1979}, p. 2).

Grassmann aspired to teach at the university level so that he could dedicate more time to his research. In the preface of his Theory of Extension of 1844, one can discern his restlessness due to his profession not allowing him to pursue scientific research, as he desired:

On one particular ground I hope to find indulgence, that the time for my research was extremely paltry and measured out piecemeal by virtue of my official position. Also my position offered me no opportunity, through reports on the domain of this science, or even related matter, to obtain that vivid freshness which must as an enlivening breath inspire the whole if it is to appear as a living limb of the organism of science. (^{GRASSMANN, 1995}, p. 17).

In 1840, Grassmann took admission exams for the Stettin Institute, which were significant for him despite the contradictions he experienced there. He had to present an essay on the theory of tides, for which he utilized the basic theory of Celestial Mechanics by Laplace and Analytical Mechanics by Lagrange, producing his essay Theory of Flow and Reflux, which introduced, for the first time, an analysis based on vectors (^{DIEUDONNÉ, 1979}).

^{Dieudonné (1979}) asserts that, although Grassmann's essay was accepted by the examiners, they did not deem the innovations introduced therein important, as his explanation was considered too abstract and philosophical. It is evident how the form of concepts was important for the mathematicians of the time, and that, lacking the academically accepted form, the content of the concepts was also disregarded. However, realizing that his theory had broad applicability, Grassmann decided to dedicate himself to the development of his ideas on vector spaces, including vector addition and subtraction, vector differentiation, and the theory of vector functions.

In 1843, Grassmann returned to Berlin and became interested in Geometry due to exams he took to improve his teaching position. According to ^{Dieudonné (1979}), Grassmann arrived at the understanding of vectors on his own, as he had not studied Mathematics beyond the high school curriculum. This reveals that Grassmann possessed a theoretical mindset and was able to establish both the internal and external connections of mathematical concepts.

Grassmann aspired to become a university professor, believing that this would facilitate the promotion of his ideas and, consequently, the dissemination of his new theory. ^{Sousa (2018}) asserts, “Human thought seeks ways to enable the continuous transformation of reality through its physical and intellectual labor [...ϡ”. He repeatedly submitted his works for evaluation by experts in the hope of obtaining a position as a university lecturer. However, they failed to comprehend him, and consequently, rejected him.

2.2 From the publication of his first work to its rewriting (1844 to 1861)

After all attempts to become a university professor were denied, Grassmann turned his attention to writing his work Die Lineale Ausdehnungslehre, ein neuer Zweig der Mathematik (^{GRASSMANN, 1878}), which was published in 1844. He created what is now called exterior algebra ⁸. However, his work did not receive much receptivity and was criticized by “[...ϡ He was mostly reproached for lack of clarity in his presentation, especially regarding the long philosophical introduction that prevented most mathematicians from reading any further.” (^{DORIER, 2000}, p. 18).

Grassmann seemed to be aware of the potential rejection his work might encounter, as he wrote:

In fact, when presenting a new science, so that its position and meaning can be duly recognized, it is essential to show at once its application and its relationship with related objects. The introduction should also serve this purpose. In the nature of things, this is more philosophical in nature, and when I separated it from the context of the entire work, I did so so as not to immediately scare mathematicians with the philosophical form. Because there is still a certain rejection among mathematicians, and partly not mistakenly, towards philosophical discussions of mathematical and physical subjects (^{GRASSMANN, 1995}, p. 15-16).

It is noted that Grassmann justified the lack of success of his work due to its presentation, which employed an unconventional language for the time, being rather philosophical and with little mathematical writing. This reflects the interdependence between the form and content of mathematical concepts, both essential for the development of this science.

In 1846, Möbius suggested, according to ^{Dieudonné (1979}), that Grassmann compete for the Fürstlich Jablonowskischen Gesellschaft prize (awarded to those who have had great relevance in the exact sciences) from the Leipzig Academy. He then submitted his book titled Die Geometrische Analyze geknüpft und die von Leibniz Characteristik, where he developed Leibniz's ideas on establishing a calculus directly applicable to geometric situations (^{OTTE, 1989}). In this study, according to Dieudonné (1979), Grassmann introduced the notion of scalar product between two vectors belonging to n-dimensional spaces. However, Möbius, who was one of the judges, criticized the way Grassmann introduced abstract ideas without providing the reader with an intuitive hook to hang them on (^{SERVIDONI, 2006}).

In May 1847, Grassmann was granted the title of Oberlehrer senior teacher) at Friedrich Wilhelm Schule, and in the same month, he wrote to the Prussian Ministry of Education requesting that his name be included on a list of those to be considered suitable for university positions. The Ministry of Education sought Kummer's opinion on Grassmann, who, through a report, dashed any hope Grassmann had of obtaining a university position, stating that the request was inadequately expressed (^{DOURADO et al., 2021}).

In early 1849, he married Therese Knappe, with whom he had eleven children, nine of whom reached adulthood (^{EVES, 2004}). In March 1852, Grassmann's father, Justus, passed away from heart failure. Later that year, Grassmann was appointed to fill his late father's former position in Stettin Gymnasium (^{DIEUDONNÉ, 1979}).

Having failed to gain recognition for his mathematics, Grassmann turned to one of his other favorite subjects, the study of Sanskrit and Gothic. Throughout his life, he gained more recognition for his language studies than for mathematics; even proving that Germanic was older in a phonological pattern than Sanskrit, establishing it as the oldest language in Indo-European linguistics. Indeed, Grassmann's remarkable versatility is evident in his dedication to numerous fields, displaying the breadth of his scientific knowledge.

In 1854, he redirected his focus to mathematics, his great passion, and his theory of extension. He was determined to rewrite his work fully, hoping to get the recognition he deserved. Simultaneously, by the year 1861, he published seventeen scientific papers that he developed for the fields of physics and mathematics. His intellectual breadth, as pointed out by ^{Servidoni (2006}), allowed him to navigate forcefully through various areas of knowledge.

3.2 From the second version of the Theory of Extension to his death (1862 to 1877)

In 1862, a new version of his 1844 work, titled Die Ausdehnungslehre, was published (^{GRASSMANN, 1862}). In this version, Grassmann defines mathematical concepts and immediately proceeds to present the applications of these concepts, thus making his ideas clearer. The alteration in the structure of the text is notable, particularly the complete omission of the philosophical introduction, as well as a methodological restructuring of the text in favor of a format closer to propositional organization, characteristic of Euclidean style. It becomes clear that, in this movement, through a process of mutability, Grassmann would be seeking to adapt his proposal to receive a positive reception, which, once again, would not be successful, at least in the 1860s (^{LINDER; SCHUBRING, 2022}). ^{Dorier (2000}) points out that, even though it no longer contained the heavy philosophical language of the first version, the reading was still dense and required meticulous study of the entire theory. This fact led the mathematicians of the time to ignore his work once again, denying him a voice within the scientific community.

Disappointed at his inability to convince mathematicians of the importance of his work, Grassmann turned once again to research in linguistics, where here, he was honored for his contributions by being elected to the American Oriental Society and awarded an honorary degree by the University of Tübingen in Germany (^{DOURADO et al., 2021}).

In 1872, Grassmann resumed his mathematical studies upon learning that mathematicians were investigating and applying his works and theories. According to his son, Justus Grassmann, two days before his passing, Grassmann received the news that his theory would be disseminated through lectures given by a professor in Dresden (GRASSMANN, 2011).

Grassmann passed away on September 26, 1877, due to cardiac problems following a period of greatly compromised health, as indicated by ^{Servidoni (2006}). ^{Dieudonné (1979}, p. 1) speaking about Grassmann, asserts that:

In the whole gallery of prominent mathematicians who, since the time of the Greeks, have left their mark on science, Hermann Grassmann certainly stands out as the most exceptional in many respects […ϡ. When compared with other mathematicians, his career is an uninterrupted succession of oddities: unusual were his studies; unusual his mathematical style; highly unusual his own belated realization of his powers as a mathematician; unusual and unfortunate the total lack of understanding of his ideas, not only during his lifetime but long after his death; deplorable the neglect which compelled him to remain all his life professor in a high-school ("Gymnasiallehrer") when far lesser men occupied University positions.

Grassmann's theory, which aligns with what Kopnin (¹⁹⁷⁸, p. 196) asserts to be a scientific theory, namely "a system, a set of judgments unified by a single principle". Although not widely accepted in his time, contained the foundations of a theory that was about to emerge in the field of Algebra, including concepts such as vector space, linear dependence, basis, dimension, and linear transformation. Due to these valuable contributions, He is widely acknowledged as the progenitor of Linear Algebra (^{ASSIS, 2018}).

According to ^{Whitehead (1898}), it is believed that Grassmann, while writing his second version of the Theory of Extension, was not even aware of Cayley's classic memoir on matrices. According to ^{Dorier (1995}), Grassmann formulated his theory independently of the rest of mathematics, relying solely on elementary rules of mathematical theoretical thought.

Grassmann's theories only gained recognition after Giuseppe Peano published a condensed version of his own interpretation of Grassmann's work, which he titled Geometric Calculus - according to the Ausdehnungslehre of H. Grassmann (^{PEANO, 2000}).

Peano, being Italian, was the most renowned mathematician of his time in his country. Despite his influence, he did not gain notoriety in his work on Geometric Calculus, likely, because he presented Grassmann's ideas rather than his own.

3.1 Vectors

According to ^{Kleiner (2007}), the earliest notions of vectors do not date back to recent centuries, nor are they related to Mathematics, but are linked to Physics and trace back to thinkers of Ancient Greece, who interpreted vectors with a "very strong geometric and intuitive appeal to aid in the analysis of physical problems" (^{TÁBOAS, 2010}, p. 2, our translation). According to Kleiner (²⁰⁰⁷) and ^{Táboas (2010}), ancient Greeks employed the concept of vectors and vector addition (nowadays known as the parallelogram rule) in the context of physical quantities such as force and velocity. These are concepts emerging from human praxis.

In Mathematics, the notion of vectors originated from the geometric representation of complex numbers, independently introduced by various authors in the late 18th and early 19th centuries. This representation was limited to points or oriented line segments on a plane. Hamilton, in his astronomical research, needing to work with rotations in the plane, found in the graphical representation of complex numbers the appropriate mathematical theory. However, he needed to relate this theory to the theory of vectors. Thus, in 1835, he algebraically defined complex numbers as ordered pairs of real numbers, possessing the usual operations of addition, multiplication, and scalar multiplication, as we have today (^{KLEINER, 2007}). In this way, Hamilton associated complex numbers with vector representation, marking the first notions of mathematical vectors and already demonstrating the interdependence of mathematical representations of concepts.

During the same period as Hamilton, Grassmann, in the second version of his work, represented a vector as a straight-line segment with specific length and direction, introducing a geometric representation for this concept, thus illustrating the mutability of conceptual notions. He defines the addition of two vectors according to the rule that we now commonly accept, and subtraction of vectors is performed as the addition of the negative, which is the vector that has the same length and opposite direction (^{MELCHIADES DA SILVA; FRANT; OLIVEIRA, 2022}).

Peano, in his interpretation of Grassmann's work, defined a vector as follows: “Every formation of the first species of the form B - A is called a vector. The points A and B are called respectively the origin and term of the vector.” (^{PEANO, 2000}, p. 30). For the author, formation of the first kind equates to a line; consequently, a formation of the form 𝐵 − 𝐴 becomes a limitation of this line, which would be, in modern language, an oriented line segment leading to the current conceptualization of a vector.

Considering the contemporary approach to the concept of vectors, ^{Boldrini et al. (1986}) state that vectors in the plane are oriented line segments with the initial point at the origin, uniquely determined by their terminal point, since the starting point is fixed at the origin. Algebraically, for each point in the plane, a unique vector is associated, and given a vector, we associate a unique point in the plane, which is its terminal point. Thus, a correspondence called biunivocal is established between points in the plane and vectors.

Extending the definition of vectors to three-dimensional space, ^{Boldrini et al. (1986}, p. 101) states that in this space there is also "a coordinate system given by three oriented lines, pairwise perpendicular, and, once a unit length is fixed, each point P in space will be indicated as a triple of real numbers (𝑥, 𝑦, 𝑧), which gives its coordinates."

It is noted that Grassmann, as expressed by ^{Peano (2000}), already conceived the core of the vector concept, and that later authors, through the historical development of the concept, refined their writing, leading to a synthesis, as developed by ^{Boldrini et al. (1986}).

3.2 Vector Space

Regarding spaces with more than two dimensions, ^{Eves (2004}) asserts that since the time Euclid published his famous work "Elements of Euclid" around 300 B.C., the Greeks already knew about geometric solids and had geometry in three dimensions. Eves (²⁰⁰⁴) reports that in Book XIII of Euclid's "Elements," the author attributes the creation of some regular solids to the Pythagoreans, suggesting the emergence of these solids centuries before the 3rd century B.C.

With the advancement of science, it became necessary to develop an algebraic theory for a space with more than two dimensions, an expansion of the existing scientific theory up to that point. After several unsuccessful attempts by various scholars to create a three-dimensional space, in 1843, Hamilton introduced quaternions, establishing a four-dimensional set; however, he lacked a solid theory for a vector space of more than two dimensions. In this context, Grassmann introduced his Theory of Extension in 1844 and 1862, proposing an extremely abstract algebra for mathematicians of the time, yet containing important traces of 𝑛 −dimensional vector analysis required by scientists. It represents the evolution of mathematical theory demanded by scholars, the fluidity of concepts.

In 1888, while elucidating Grassmann's ideas, Peano introduced his linear systems, which we now know, in modern terms, as vector spaces. He stated that a linear system is a system of entities that satisfy the conditions 1, 2, 3, and 4 below:

Let there exist a system of entities for which are given the following definitions:

1. The equivalence of two entities 𝑎 and 𝑏 of the system is defined, that is, a proposition, indicated by 𝑎 = 𝑏, is defined, which expresses a condition between two entities of the system, satisfied by certain pairs of entities, and not by others, and which satisfies the logical equations:

(𝑎 = 𝑏) = (𝑏 = 𝑎), (𝑎 = 𝑏) ∩ (𝑏 = 𝑐) < (𝑎 = 𝑐)

2. The sum of two entities 𝑎 and 𝑏 is defined, that is to say an entity, indicated by 𝑎 + 𝑏, is defined that also belongs to the system given, and which satisfies the conditions

(𝑎 = 𝑏) < (𝑎 + 𝑐 = 𝑏 + 𝑐), 𝑎 + 𝑏 = 𝑏 + 𝑎, 𝑎 + (𝑏 + 𝑐) = (𝑎 + 𝑏) + 𝑐, the common value of the two members of the last equivalence being indicated by 𝑎 + 𝑏 + 𝑐.

3. If 𝑎 is an entity of the system, and m is ?? positive integer, by the

expression 𝑚𝑎 we will mean the sum of 𝑚 entities equal to 𝑎. It is easy to recognize that, if 𝑎, 𝑏, … are entities of the system, and 𝑚, 𝑛, … positive integers,

(𝑎 = 𝑏) < (𝑚𝑎 = 𝑚𝑏); 𝑚(𝑎 + 𝑏) = 𝑚𝑎 + 𝑚𝑏; (𝑚 + 𝑛)𝑎 = 𝑚𝑎 + 𝑛𝑎; 𝑚(𝑛𝑎) = (𝑚𝑛)𝑎; 1𝑎 = 𝑎.

We will suppose that a meaning is attributed to the expression rna, whatever may be the real number 𝑚, in such a way that the preceding equations are also satisfied. The entity 𝑚𝑎 is called the product of the (real) number 𝑚 with the entity 𝑎.

4. Finally we will suppose there exists an entity of the system, which we will call the null entity, and that we will indicate by 0, such that whatever may be the entity 𝑎, the product of the number 0 with the entity 𝑎 always gives the entity 0, that is 0𝑎 = 0. If to the expression 𝑎 − 𝑏 one attributes the meaning 𝑎 + (−1)𝑏, one deduces

𝑎 − 𝑎 = 0, 𝑎 + 0 = 𝑎 (PEANO, 1888, p. 119-120).

^{Dorier (2000}) clarifies that while Grassmann deduced the fundamental properties of vector spaces from the definition of operations on coordinates, Peano was more precise and described an axiomatic structure from which they emerged. Peano also refined the formulation by removing some redundancies and providing greater clarity to the concepts of zero and opposite elements. Thus, Peano went a step further than Grassmann, contributing to the improvement and development of the concept of vector space, leading to a mutation of ideas through established judgment, towards a relatively finished idea (KOPNIN, 1978), following his studies of the Theory of Extension.

With the contribution of other scholars, the concept continued to develop, transform, and its expression improved until reaching the synthesis we have today, as can be expressed according to ^{Boldrini et al. (1986}), defining a vector space as a non-empty set 𝑉 that possesses addition and scalar multiplication operations, given, respectively, by:

+: 𝑉 × 𝑉 → 𝑉 ∙: 𝑅 × 𝑉 → 𝑉

(𝑢, 𝑣) ↦ 𝑢 + 𝑣 (𝑎, 𝑢) ↦ 𝑎𝑢

such that, for any 𝑢, 𝑣, 𝑤 ∈ 𝑉 and 𝑎, 𝑏 ∈ 𝑅, the following properties are satisfied:

𝑖(𝑢 + 𝑣) + 𝑤 = 𝑢 + (𝑣 + 𝑤)

𝑖𝑖𝑢 + 𝑣 = 𝑣 + 𝑢

𝑖𝑖𝑖There exists 0 ∈ 𝑉 such that 𝑢 + 0 = 𝑢.(Here,0it is called the zero vector.)

𝑖𝑣There exists - 𝑢 ∈ 𝑉 such that 𝑢 + (−𝑢) = 0

𝑣𝑎(𝑢 + 𝑣) = 𝑎𝑢 + 𝑎𝑣

𝑣𝑖(𝑎 + 𝑏)𝑣 = 𝑎𝑣 + 𝑏𝑣

𝑣𝑖𝑖(𝑎𝑏)𝑣 = 𝑎(𝑏𝑣)

𝑣𝑖𝑖𝑖1𝑢 = 𝑢 (^{BOLDRINI et al., 1986}, p. 103, our translation).

With this definition, the richness of Peano's contributions to the axiomatization of Mathematics is evident, as through this axiomatization, it was possible to establish clear and concise syntheses of mathematical concepts such as that of a vector space.

3.3 Linear Dependence and Independence

Grassmann, according to ^{Táboas (2010}), did not originally conceive the concepts of linear dependence and independence. However, the author states that Grassmann, in his Theory of Extension, presented significant aspects for the organization and formalization of these concepts in axiomatic terms, placing them in a broader context within Mathematics.

Grassmann approached the definition of linear dependence by stating: "We say that an elementary quantity of the first degree is dependent on other elementary quantities when it can be represented by a linear combination of these latter" (^{MELCHIADES DA SILVA; FRANT; OLIVEIRA, 2022}, p. 20-21, our translation).

Grassmann also referred to the concept of dependence as one species being dependent on another species, when a vector could be written as a linear combination of other vectors (^{MELCHIADES DA SILVA; FRANT; OLIVEIRA, 2022}), exactly as we have it today.

We also find the notion of linear dependence expressed in the following terms: "One kind of change (Aenderungsweise) is dependent on another when the vectors of the first can be represented as a sum of vectors of the second [...ϡ” (^{MELCHIADES DA SILVA; FRANT; OLIVEIRA, 2022}, p. 20, our translation).

However, according to the author, this statement is not sufficient to decide when the given vectors are linearly independent or, in Grassmann's terms, independent of each other, as there are still inherent contradictions in it.

^{Caire (2020}) also highlights a notion of linear dependence presented in the context of what Grassmann referred to as magnitudes:

[Grassmannϡ defined the concept of dependence, where an elementary magnitude of the first order was represented as a multiple sum of independent elementary magnitudes of the first order, and these independent magnitudes could not be represented as a multiple or sum of the remaining independent magnitudes (^{CAIRE, 2020}, p. 80, our translation).

Through Peano, we see an improvement in the mathematical terms expressed by Grassmann, with the elimination of conceptual contradictions. In Chapter IX of his "Geometric Calculus", Peano provides the following definition of linearly dependent and independent.

Several entities 𝑎1, 𝑎2, … 𝑎𝑛of a linear system are called mutually dependent if one can determine 𝑛 numbers 𝑚1, 𝑚2, … 𝑚𝑛, not all zero, for which there results exist a relation

𝑚1𝑎1 + ⋯ + 𝑚𝑛𝑎𝑛 = 0.

In this case anyone of the entities whose coefficient is not zero can be expressed as a linear homogeneous function of the rest. If the entities 𝑎₁ … 𝑎𝑛 are mutually independent, and if between them there is a relation𝑚1𝑎1 + ⋯ + 𝑚𝑛𝑎𝑛, one deduces 𝑚1 = 0, … , 𝑚𝑛 = 0. If 𝐴𝐵 … are formations of first species in space, the equations 𝐴𝐵 = 0, 𝐴𝐵𝐶 = 𝑂, 𝐴𝐵𝐶𝐷 = 0 express the dependence of 2 or 3 or 4 formations. 5 formations of the first species in space are always mutually dependent (PEANO, 1888, p.120).

Comparing Peano's language with that used in contemporary textbooks, it is noted that the term "mutually" is equivalent to the term "linearly." ^{Boldrini et al. (1986}) state that it is of utmost importance to know whether a vector is or is not a linear combination of others, and they provide a synthesis of the entire historical development of the concepts of linearly dependent and linearly independent, expressed as follows:

Let 𝑉 be a vector space and 𝑣₁, … , 𝑣𝑛 ∈ 𝑉. We say that the set

{𝑣1, … , 𝑣𝑛} is linearly independent (LI), or that the vectors {𝑣1, … , 𝑣𝑛}

are LI, if the equation 𝑎1𝑣1 + ⋯ = 𝑎𝑛𝑣𝑛 = 0 implies that 𝑎1 = 𝑎2 =

⋯ = 𝑎𝑛 = 0. In the case where there exists some 𝑎?? ≠ 0, we say that

{𝑣1, … , 𝑣𝑛} is linearly dependent (LD), or that the vectors 𝑣1, … , 𝑣𝑛

are LD (^{BOLDRINI et al., 1986}, p. 114, emphasis added).

The author further presents a result, in the form of a theorem, to clarify and classify linearly dependent vectors: “{𝑣₁, … , 𝑣𝑛} It is LD if, and only if, one of these vectors is a linear combination of the others” (^{Boldrini et al., 1986}, p. 114). The resemblance of the current notation to the notation introduced by Peano almost a hundred years before the work of Boldrini et al. (1986) is noticeable, reaffirming that the evolution of this concept manifests through the movement of the history of its formation and development (^{SOUSA, 2018}).

3.4 Base and dimension

^{Crowe (1967}), analyzing the works of Grassmann, brings the following result, which he says is a typical assertion of Grassmann: “Every vector of a system of the 𝑚^𝑡ℎ order can be expressed as the sun of m given independet manners of change of the system. This expression is unique (^{CROWE, 1967}, p. 69).” (^{Crowe, 1967}, p. 69). In this passage, the concept of the dimension of a vector space is introduced when he speaks of a system of order 𝑚, and when he says "the sum of 𝑚 vectors belonging to the given 𝑚 independent ways of changing the system," he is referring to a linear combination of vectors that are linearly independent. Combining these two parts, it can be affirmed that Grassmann was referring to the basis of a vector space, which is a set of linearly independent vectors that span the entire space. Moreover, being spanned implies being written as a linear combination of some given vectors; consequently, these given vectors are the elements of the basis, whose quantity represents the dimension of the space.

Since the initial representations of the concepts of basis and dimension, their interdependence with other concepts such as linear combination and linear dependence is evident, demonstrating that the concepts are established within a conceptual system that requires theoretical thinking and specific mathematical forms of thinking that relate the concepts in order to develop new concepts.

From a contemporary perspective, this result can be rewritten as: Every vector in a vector space of dimension 𝑚 can be written as a linear combination of 𝑚 linearly independent vectors. This result resembles what ^{Boldrini et al. (1986}, p. 120, our translation) present as a theorem, which states: “Given a basis 𝛽 = {𝑣₁, 𝑣₂, … , 𝑣𝑛} of 𝑉, each vector in 𝑉 is uniquely written as a linear combination of 𝑣1, 𝑣2, … , 𝑣𝑛.”

^{Peano (2000}, p. 120), refining Grassmann's language, in defining dimension, writes that "The number of dimensions of a linear system is the maximum number of mutually independent entities of the system that one can take." To make his definition clearer, he provides an example:

For example, the formations of the first species on a right-line, or in the plane, or in space, form linear systems of 2, 3, and 4 dimensions, respectively; the vectors in the plane or in space form systems of 2 and 3 dimensions; the formations of the second species in space form a system of 6 dimensions. The real numbers form a linear system of one dimension; the imaginary numbers, or ordinary complexes, form a system of two dimensions. A linear system can also have infinite dimensions (^{PEANO, 1888}, p.120).

It is evident that Peano refers to the existence of vector spaces of infinite dimension; consequently, the notion of basis is implicit in this concept through its internal connections.

Formalizing all these concepts, ^{Boldrini et al. (1986}) writes that a set {𝑣1, … , 𝑣𝑛} of vectors in a vector space 𝑉 will be a basis of 𝑉 if they are linearly independent and span the entire space. Moreover, "Any basis of a vector space always has the same number of elements. This number is called the dimension of 𝑉, denoted 𝑑𝑖𝑚𝑉." (^{BOLDRINI et al., 1986}, p. 119, author's emphasis). Thus, we have a synthesis of the concepts initially presented by Grassmann and refined by Peano and other authors who, through the development of history, contributed to the improvement of mathematical notation.

3.5 Linear Transformation

In the 1862 version of the Theory of Extension, Grassmann added a part that was not present in the 1844 version and named it Funktionenlehre (Theory of Functions), which, according to Liesen (2011), it is the most splendid addition made by Grassmann. It is in this part that we find a section entitled "Ganze Funktionen ersten Grades und Darstellung derfelben als Quotienten" (Complete functions of the first degree and their representation as quotients), where Grassmann works with the mathematical object called "Bruch" or "Quotient" (Fraction or Quotient). In modern terminology, Grassmann's "fraction" can be referred to as a linear transformation of a finite-dimensional vector space (^{LIESEN, 2011}).

^{Peano (2000}), in representing history through logic via his interpretation of Grassmann, rewrites the definition of linear transformation as follows:

An operation𝑅, to be carried out on every entity a of a linear system𝐴, is called distributive, if the result of the operation 𝑅 on the entity a, which we will indicate by 𝑅𝑎, is also an entity of a linear system, and the identities

𝑅(𝑎 + 𝑎′) = 𝑅𝑎 + 𝑅𝑎′, 𝑅(𝑚𝑎) = 𝑚(𝑅𝑎),

are verified, where 𝑎 and 𝑎′ are any entities whatever of the system

𝐴, and 𝑚 is any real number. The entity𝑅𝑎, that is the result of the distributive operation 𝑅 on the entity 𝑎, is called a distributive function of 𝑎. A distributive operation is also called a linear transformation, or transformation without any qualifier (PEANO, 1888, p. 122).

Through a temporal movement, bringing the concept of linear transformation into the present day, ^{Boldrini et al. (1986}, p. 144, author's emphasis) write:

Let 𝑉 and 𝑊 be two vector spaces. A linear transformation (linear mapping) is a function from 𝑉 to𝑊, denoted as 𝐹: 𝑉 → 𝑊, that satisfies the following conditions:

i) For any vectors 𝑢 and 𝑣 in𝑉, 𝐹(𝑢 + 𝑣) = 𝐹(𝑢) + ??(𝑣),

𝑖𝑖For any scalar 𝑘 ∈ 𝑅and vector ∈ 𝑉, 𝐹(𝑘𝑣) = 𝑘𝐹(𝑣).

It is notable that both works define linear transformation in the same manner, despite the different symbols used, indicating that in Peano's time the concept was already forming and that, with history, there was a mutation in mathematical symbolic writing.