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# Linear map (linear operator)

| 🗖 nosplit | ↑ parent "Linear algebra" | words: 201 | descendant words: 1k | descendants: 21
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Every linear map in finite dimension can be represented by a square matrix, the points of the domain being represented as vectors.
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As such, when we say "linear map", we can think of a generalization of matrix multiplication that makes sense in infinite dimensional spaces like Hilbert spaces, since calling such infinite dimensional maps "matrices" is stretching it a bit, since we would need to specify infinitely many rows and columns.
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The prototypical building block of infinite dimensional linear map is the derivative. In that case, the vectors being operated upon are functions, which cannot therefore be specified by a finite number of parameters, e.g.
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For example, the left side of the time-independent Schrodinger equation is a linear map. And the time-independent Schrodinger equation can be seen as a eigenvalue problem.
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