[Verse 1]
Picture dimensions flowing through a gate
Some survive the journey, some meet their fate
In this mathematical dance of space and time
Nothing vanishes without reason or rhyme
[Chorus]
Kernel plus image equals n
What gets killed plus what remains
Nullity plus rank, the columns count the same
Conservation law that never breaks the chain
Dim of ker plus dim of im
Always adds up to where we begin
[Verse 2]
Five dimensions enter, two collapse and fall
Three emerge victorious, standing proud and tall
The kernel holds the casualties, the shadow's hidden cost
While the image shows us everything that wasn't lost
[Chorus]
Kernel plus image equals n
What gets killed plus what remains
Nullity plus rank, the columns count the same
Conservation law that never breaks the chain
Dim of ker plus dim of im
Always adds up to where we begin
[Bridge]
When Ax equals b has solutions to find
The answer set's a translate, geometrically aligned
With kernel's dimension, that's the freedom we possess
The theorem tells us exactly what to expect
[Verse 3]
Three-dimensional world mapped down to two
One dimension vanishes, but the math stays true
That missing piece becomes the kernel's home
While the image captures what survived the roam
[Final Chorus]
Kernel plus image equals n
What gets killed plus what remains
Nullity plus rank, the columns count the same
Conservation law that never breaks the chain
Dim of ker plus dim of im
Always adds up to where we begin
[Outro]
Nothing created, nothing truly dies
Just redistributed before our eyes
The Rank-Nullity Theorem stands
Dimensions balanced in its hands