[Verse 1] When systems call your name, Ax equals b Row reduce the matrix, set the variables free Augmented form beside you, pivot through each row Gaussian elimination, watch the answer grow [Chorus] Eleven sacred formulas, carved in algebraic stone Dimension, determinant, eigenvalues we own From Cauchy-Schwarz to spectral, SVD's embrace These equations hold the keys to linear vector space [Verse 2] Dimension tells the story of how big your space can be Count the basis vectors, that's your dim of V U plus W together, minus intersection's share Rank-nullity theorem, kernel plus image there [Chorus] Eleven sacred formulas, carved in algebraic stone Dimension, determinant, eigenvalues we own From Cauchy-Schwarz to spectral, SVD's embrace These equations hold the keys to linear vector space [Verse 3] Determinant's got power, AB product rule Det of A times det B, multiplication's tool Eigenvalues multiply to give you det complete Volume scaling factor, geometric feat [Bridge] Lambda minus A times I, set determinant to zero Find your eigenvalues, be your matrix hero Trace equals sum of lambdas, product gives det Cayley-Hamilton whispers, p of A you bet [Verse 4] Inner products dancing, Cauchy-Schwarz won't lie Absolute of u dot v, bounded by their size Projection formula casting shadows on the wall Parseval's identity, orthonormal call [Chorus] Eleven sacred formulas, carved in algebraic stone Dimension, determinant, eigenvalues we own From Cauchy-Schwarz to spectral, SVD's embrace These equations hold the keys to linear vector space [Outro] Symmetric matrices singing Q D Q transpose SVD exists for all, decomposition close U sigma V transpose, the universal way Eleven formulas guide us through algebra's maze
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