[Verse 1] Don't think of vectors as arrows in space They're elements living in a vector place Functions and matrices, polynomials too All can be vectors, that's something brand new The space defines them, not the way they point Abstract algebra is the joining point [Chorus] Seven foundations, seven keys to see Linear transformations, that's our destiny Eigenvalues show us, rank reveals the way Inner products guide us, duality at play Basis is our language, geometry's the game Seven keys unlock it, algebra's true name [Verse 2] Linear transformation IS its matrix form Once you pick your bases, they become the norm Don't think they're different, don't think they're apart The matrix IS the mapping, same mathematical heart Two views of one object, like faces of a coin Where computation and abstraction join [Chorus] Seven foundations, seven keys to see Linear transformations, that's our destiny Eigenvalues show us, rank reveals the way Inner products guide us, duality at play Basis is our language, geometry's the game Seven keys unlock it, algebra's true name [Verse 3] Eigenvalues reveal what's truly there The intrinsic geometry beyond compare In the eigenbasis everything just scales Natural coordinates tell the deeper tales Find the eigenvalues, find the truth inside Where the transformation cannot hide [Bridge] Choice of basis, choice of tongue to speak Same reality, different technique Some make computation smooth and clean Spectral theorem finds the best machine [Verse 4] Inner products bring geometry to life Orthogonality cuts through like a knife Projection and least squares, all from one source The inner product shows us our true course Rank measures information, how much gets through Tells you what the transformation really can do [Chorus] Seven foundations, seven keys to see Linear transformations, that's our destiny Eigenvalues show us, rank reveals the way Inner products guide us, duality at play Basis is our language, geometry's the game Seven keys unlock it, algebra's true name [Outro] Duality whispers, every space has shade V-star measures what V has made V-double-star brings us home again Objects and measures, the eternal chain Seven keys together, unlock the door To algebra's kingdom, and so much more
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