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Unformatted text preview: 1.6 Homogeneous Systems, Subspaces and Bases 1 Chapter 1. Vectors, Matrices, and Linear Spaces 1.6. Homogeneous Systems, Subspaces and Bases Definition. A linear system A~x = ~ b is homogeneous if ~ b = ~ 0. The zero vector ~x = ~ 0 is a trivial solution to the homogeneous system A~x = ~ 0. Nonzero solutions to A~x = ~ 0 are called nontrivial solutions. Theorem 1.13. Structure of the Solution Set of A~x = ~ . Let A~x = ~ 0 be a homogeneous linear system. If ~ h 1 , ~ h 2 ,..., ~ h n are solu- tions, then any linear combination r 1 ~ h 1 + r 2 ~ h 2 + + r n ~ h n is also a solution. Proof. Since ~ h 1 , ~ h 2 ,..., ~ h n are solutions, A ~ h 1 = A ~ h 2 = = A ~ h n = ~ and so A ( r 1 ~ h 1 + r 2 ~ h 2 + + r n ~ h n ) = r 1 A ~ h 1 + r 2 A ~ h 2 + + r n A ~ h n = ~ 0+ ~ 0+ + ~ 0 = ~ . Therefore the linear combination is also a solution. QED 1.6 Homogeneous Systems, Subspaces and Bases 2 Definition 1.16. A subset W of R n is closed under vector addition if for all ~u,~v W , we have ~u + ~v W . If r~v W for all ~v W and for all r R , then W is closed under scalar multiplication . A nonempty subset W of R n is a subspace of R n if it is both closed under vector addition and scalar multiplication....
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