[Verse 1] Not just arrows flying through the air Vectors live in spaces everywhere Polynomials dancing, functions singing Matrices spinning, all belonging Fields and sets with operations two Addition pairs with scaling through Eight conditions mark the sacred ground Where vector spaces can be found [Chorus] Abelian group for addition's sake Closure, inverse, zero for the break Scalar multiplication must distribute clean Compatible with fields, the smoothest machine One times vector equals itself so true Vector spaces hold more than me and you [Verse 2] Real numbers lined in perfect rows Continuous functions as the interval flows Solutions to equations homogeneous Even zero standing lonesome Every polynomial of degree three Lives inside this family The mindset shifts from arrows straight To elements that calculate [Chorus] Abelian group for addition's sake Closure, inverse, zero for the break Scalar multiplication must distribute clean Compatible with fields, the smoothest machine One times vector equals itself so true Vector spaces hold more than me and you [Bridge] Subspaces nested like Russian dolls Three conditions guard the walls Zero element must reside Addition stays locked inside Scalar products cannot flee Closure keeps the family [Verse 3] Test your subset for the crown Zero lives there, written down Sum two vectors, stay within Scale by constants, always win From the trivial space alone To infinite dimensions grown Every element's a vector now Polynomial, function, matrix row [Chorus] Abelian group for addition's sake Closure, inverse, zero for the break Scalar multiplication must distribute clean Compatible with fields, the smoothest machine One times vector equals itself so true Vector spaces hold more than me and you [Outro] Eight axioms standing guard Making vector spaces starred Not just arrows anymore Welcome to the abstract shore
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