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Definition: A subspace of R^n is any collection S of vectors in R^n such that: 1) The zero vector is in S. 2) If u and v are in S, then u + v is in S. (S is closed under addition.) 3) If u is in S and c is a scalar, then cu is in S. (S is closed under scalar multiplication.)
Source: Linear Algebra: A Modern Introduction, 3rd edition by David Poole (note-custom edition titled Matrix Algebra)