## Vector spaces, Linear transformations |

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automorphism basis vectors codomain coefficients column complete inverse image complex numbers coordinates definition dependent Differential Equations dimension theorem Equation 2.2 Euclidean Spaces examples of vector Exercise field F finite set finite-dimensional vector space Foundation Course function geometric vectors Hence Hermite Normal Form introduced isomorphism Jordan Normal Form kernel Linear Algebra linear combination Linear Mathematics linear relation linear transformation linearly independent set linearly independent subset look mapping morphism n-dimensional vector space n-tuples notation Notes number of elements one-one one-to-one page N12 page N27 paragraph polynomials of degree proof prove Rank and Nullity reading passage real numbers real vector space replacement process Replacement Theorem representation scalar multiplication Self-assessment Test set of vectors set spanned set which spans spanning set spans R2 statement of Theorem sub-section subspace Summary of Section taking of linear Theorem 3.1 transformation page N31 unique vectors in R2 xu x2 zero polynomial zero vector