High Entropy Thoughts

Early on we learn transpose of a matrix as a new matrix formed by interchanging the rows and columns of the original matrix. Matrix transpose isn't just a mechanical row-column interexchange but carries a deeper conceptual meaning—one tied to duality and linear functionals in linear algebra.

Let me take you on a journey from vector spaces to dual spaces, using a plumbing analogy that helped me think through these concepts.

Imagine a water distribution system. We have faucets that we can turn on with different intensities and buckets that collect the flowing water. Between the faucets and buckets we have a network of pipes, each with specific width that determines how much water flows through. Lets represent this pipe network as a matrix as shown below

Linear Map: A linear map is a formally defined as T: V → W, where T transforms vectors from vector space V to vector space W while preserving linear structure. Basically a linear function generalized to vector spaces.

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