@misc{VECTSP13.MIZ,
      AUTHOR = {Schwarzweller, Christoph and Rowi\'{n}ska-Schwarzweller, Agnieszka},
      TITLE = {Some Standard Examples of Vector Spaces}, 
      ADDRESS1 = {Institute of Informatics, University of Gda{\'n}sk, Poland},
      ADDRESS2 = {Institute of Informatics, University of Gda{\'n}sk, Poland},
      SECTION1 = {Preliminaries},
      SECTION2 = {On Linear Transformations},
      SECTION3 = {The Vector Space $R^n$},
      SECTION4 = {The Vector Space of Maps into a Ring $R$},
      SECTION5 = {The Vector Space of Linear Transformations},
      DAY = {24},
      MONTH = {December},
      YEAR = {2024},
      SUMMARY = {In this article we introduce some standard vector spaces \cite{Lang2002}: the vector
                 space of $n$-tuples over a commutative ring $R$, the vector space of
                 of maps from a set $X$ into a ring $R$, and the vector space of
                 linear transformations between vector spaces.
                 We prove that two finite-dimensional vector spaces are isomorphic if 
                 and only if they are of the same dimension, and hence each vector space of 
                 dimension $n$ is isomorphic to a vector space of $n$-tuples \cite{Isaacs:2009},
		 \cite{Christmann:1989}.
                 In the course of proving the above we also showed, that linear transformations 
                 are uniquely determined by their values with respect to the basis.}}

@BOOK{Isaacs:2009,
      AUTHOR = {Isaacs, I.~Martin},
      TITLE = {Algebra, {A} {G}raduate {C}ourse},
      PUBLISHER = {American Mathematical Society},
      YEAR = {2009}}

@BOOK{Christmann:1989,
      AUTHOR = {Christmann, Norbert},
      TITLE = {Einf\"{u}hrung in die (lineare) {A}lgebra},
      PUBLISHER = {Lecture Notes, University of Kaiserslautern, Germany},
      YEAR = {1989}}