[Verse 1]
In the algebra theater where transformations dance
Every vector plays a role in this linear romance
Some get mapped to zero, trapped in kernel's embrace
Others venture through the door to image's bright space
[Chorus]
Rank-nullity sings the conservation song
Dimension in equals dimension out plus destroyed along
What goes in either comes out or gets lost in the void
The total stays constant, nothing's been deployed
Spectral theorem shows symmetric truth
Real eigenvalues, orthogonal proof
Cauchy-Schwarz bounds the projection game
Shadow never grows taller than the frame
[Verse 2]
Symmetric matrices wear no disguise or mask
Real eigenvalues answer when you ask
Take conjugate transpose of that eigenvector line
Watch complex numbers vanish, leaving real behind
[Chorus]
Rank-nullity sings the conservation song
Dimension in equals dimension out plus destroyed along
What goes in either comes out or gets lost in the void
The total stays constant, nothing's been deployed
Spectral theorem shows symmetric truth
Real eigenvalues, orthogonal proof
Cauchy-Schwarz bounds the projection game
Shadow never grows taller than the frame
[Bridge]
Extend kernel basis to the whole domain
Image basis formed from what remains
Perpendicular stretching, crystal clear
No hidden twists when symmetry's here
[Verse 3]
Inner product bounded by the norm duet
Expand that quadratic, discriminant in debt
Length times length creates the ceiling height
Projection's shadow stays within the light
[Chorus]
Rank-nullity sings the conservation song
Dimension in equals dimension out plus destroyed along
What goes in either comes out or gets lost in the void
The total stays constant, nothing's been deployed
Spectral theorem shows symmetric truth
Real eigenvalues, orthogonal proof
Cauchy-Schwarz bounds the projection game
Shadow never grows taller than the frame
[Outro]
Balance sheets and stretching axes clean
Universal bounds keep projections lean
Three pillars standing in algebra's hall
Conservation, symmetry, and bounded call