[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