# c1s6 - 1.6 Homogeneous Systems Subspaces and Bases 1...

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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 Ax = b is homogeneous if b = 0. The zero vector x = 0 is a trivial solution to the homogeneous system Ax = 0. Nonzero solutions to Ax = 0 are called nontrivial solutions. Theorem 1.13. Structure of the Solution Set of Ax = 0 . Let Ax = 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, Ah 1 = Ah 2 = · · · = Ah n = 0 and so A ( r 1 h 1 + r 2 h 2 + · · · + r n h n ) = r 1 Ah 1 + r 2 Ah 2 + · · · + r n Ah n = 0+0+ · · · +0 = 0 . Therefore the linear combination is also a solution. QED

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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 rv 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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