[Verse 1] When functions travel space to space T maps V to W with perfect grace Addition stays addition, scaling holds its ground Two rules to check, then linearity's found T of u plus v equals T-u plus T-v And c times input gives c times what you see [Chorus] Linear transformations, bend but never break Matrix representations, every choice we make Kernel kills to zero, image shows the range Rank plus nullity, dimensions rearrange Conservation laws in abstract algebra's domain [Verse 2] Rotation spins the plane around Reflection flips what can be found Derivatives drop polynomial degree Integration sums from A to B Matrix multiplication, Ax transforms the vector Every linear map needs its connector [Chorus] Linear transformations, bend but never break Matrix representations, every choice we make Kernel kills to zero, image shows the range Rank plus nullity, dimensions rearrange Conservation laws in abstract algebra's domain [Bridge] Pick your bases, build the bridge Column J holds T of V-J in the ridge What gets killed forms the kernel space What survives shows the image face Nullity plus rank equals the source dimension Information tax, no intervention [Verse 3] Five-D input, three-D output means Two dimensions lost behind the scenes The kernel catches what maps to naught The image holds what can be brought Nothing vanishes without a trace Everything accounted in this mathematical space [Final Chorus] Linear transformations, bend but never break Matrix representations, every choice we make Kernel kills to zero, image shows the range Rank plus nullity, dimensions rearrange Conservation laws in abstract algebra's domain T preserves the structure, keeps the rules the same
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