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  "body": "\n## Summary\n\nA representation of a [[quiver|RE9YFFS3TS]] over a field assigns a [[vector space|000AYPA9AJ]] to each vertex.\nIt also assigns to each arrow a [[linear map|0NRYHJPQG2]] from the space at its source to the space at its target.\nRepresentations of a finite quiver are the same as modules over its [[path algebra|2PYANBGXFD]].\nClassifying the [[indecomposable|E2P29GCD9T]] representations is a central problem; [[Gabriel's theorem|FC78F2E87E]] and [[Kac's theorem|59QJC3ZXA1]] describe their dimension vectors through root systems.\nMany problems of linear algebra, such as the classification of pairs of matrices, are problems about quiver representations.\n\n## Explanations\n\n### L1 human\n\nTake a quiver, a diagram of dots and arrows.\nPut a vector space at each dot and a matrix along each arrow, going from the space at the start to the space at the end.\nThat is a representation.\nFor the quiver with one vertex and one loop, a representation is a single square matrix.\nOver the complex numbers its classification up to change of basis is the Jordan normal form.\nFor one arrow $1 \\to 2$, a representation is a linear map, determined up to isomorphism by the two dimensions and its rank.\n",
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