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Definition: Let V be a set on which two operations, called addition and scalar multiplication, have been defined. If the vectors u and v are in V, the sum of u and v is denoted by u+v, and if c is a scalar, the scalar multiple of u by c is denoted by cu. If the following axioms hold for all u, v, and w in V and for all scalars c and d, then V is called a vector space and its elements are called vectors. 1) u+v is in V. 2) u+v = v+u. 3) (u+v)+w = u+(v+w). 4) There exists an element 0 in V, called a zero vector, such that u+0 = u. 5) For each u in V, there is an element -u in V such that u+(-u) = 0. 6) cu is in V. 7) c(u+v) = cu+cv. 8) (c+d)u = cu+du. 9) c(du) = (cd)u. 10) 1u = u.
Source: Linear Algebra: A Modern Introduction, 3rd edition by David Poole (note-custom edition titled Matrix Algebra)