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

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

Page1 / 6

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online