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Once we have vectors, the next most important abstraction in linear algebra is the concept of a linear mapping from one vector space to another. Since we are concerned only with vectors of finite dimension, such a mapping can always be represented using a matrix of coefficients. Ax where A is a matrix, and Ax is an inner product. Often, the matrix A has some special structure, for example it may be banded or symmetric. Each type has as similar an interface as possible, so once you've learned one matrix type, learning others is easy.