[Verse 1] Picture shadows dancing on the wall When sunlight hits at angle's call Your vector casts its ghostly twin Upon the space it's landing in The projection shows what can be caught While perpendicular parts are lost [Chorus] Vector dot vector over vector dot itself Times the vector makes the shadow's stealth Projection equals closest point you'll find Orthogonal parts get left behind V equals projection plus V-perp Shadow plus the leftover reserves [Verse 2] Any basis set can be refined Through Gram-Schmidt's algorithmic mind Take each vector, subtract away All projections from the previous day Normalize to unit length Build orthonormal strength [Chorus] Vector dot vector over vector dot itself Times the vector makes the shadow's stealth Projection equals closest point you'll find Orthogonal parts get left behind V equals projection plus V-perp Shadow plus the leftover reserves [Bridge] When data points scatter wild and free Linear regression holds the key Project onto the line that fits Minimize the error's twists Least squares makes the residuals dance Perpendicular to your best chance [Verse 3] Subspace W becomes your stage Vector V performs with sage Proj-W-V approximates best While V-perpendicular protests This decomposition theorem shows How every vector's story goes [Chorus] Vector dot vector over vector dot itself Times the vector makes the shadow's stealth Projection equals closest point you'll find Orthogonal parts get left behind V equals projection plus V-perp Shadow plus the leftover reserves [Outro] Shadows tell us what we can retain Errors whisper what we can't explain Orthogonal decomposition's art Splits every vector's beating heart
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