Lect16 - Chapter 3. Vector Spaces Math1111 Basis and...

Info iconThis preview shows pages 1–10. 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

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

View Full DocumentRight Arrow Icon

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

View Full DocumentRight Arrow Icon

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

View Full DocumentRight 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: Chapter 3. Vector Spaces Math1111 Basis and Dimension Examples (Cont’d) Example . Let U and V be subspaces of a vector space. If U ∩ V = { } , show that dim ( U + V ) = dim U + dim V . Proof . Let dim U = m and { u 1 , ··· , u m } be a basis of U . Let dim V = n and { v 1 , ··· , v n } be a basis of V . Want to show: u 1 , ··· , u m , v 1 , ··· , v n form a basis for U + V . Definition When U ∩ V = { } , the subspace U + V is called the direct sum of U and V , denoted by U ⊕ V . Chapter 3. Vector Spaces Math1111 Basis and Dimension Examples (Cont’d) Theorem ( Dimension Theorem ) Let U and V be subspaces of a vector space. Then dim ( U + V ) = dim U + dim V- dim ( U ∩ V ) . Chapter 3. Vector Spaces Math1111 Basis and Dimension Examples (Cont’d) Theorem ( Dimension Theorem ) Let U and V be subspaces of a vector space. Then dim ( U + V ) = dim U + dim V- dim ( U ∩ V ) . Remark dim ( U ⊕ V ) = dim U + dim V Chapter 3. Vector Spaces Math1111 Basis and Dimension Examples (Cont’d) Example . Find dim N ( A ) if A = 1 2 1 2 1 2 1 1 1 1 1 . Chapter 3. Vector Spaces Math1111 Basis and Dimension Examples (Cont’d) Example . Find dim N ( A ) if A = 1 2 1 2 1 2 1 1 1 1 1 . Ans . In this case, N ( A ) = { x ∈ R 5 : A x = } The solutions of A x = are x = α u + β v , where u = (- 1 1 ) T and v = (- 1 1 1 ) T . Chapter 3. Vector Spaces Math1111 Basis and Dimension Examples (Cont’d) Example . Find dim N ( A ) if A = 1 2 1 2 1 2 1 1 1 1 1 . Ans . In this case, N ( A ) = { x ∈ R 5 : A x = } The solutions of A x = are x = α u + β v , where u = (- 1 1 ) T and v = (- 1 1 1 ) T . Apply elementary row operations: 1 2 1 2 1 2 1 1 1 1 1 -→······-→ 1 1 1 1 1- 1 Chapter 3. Vector Spaces Math1111 Basis and Dimension Examples (Cont’d) Example . Find dim N ( A ) if A = 1 2 1 2 1 2 1 1 1 1 1 . Ans . In this case, N ( A ) = { x ∈ R 5 : A x = } The solutions of A x = are x = α u + β v , where u = (- 1 1 ) T and v = (- 1 1 1 ) T . ∴ N ( A ) = Span ( u , v ) . Chapter 3. Vector Spaces Math1111 Basis and Dimension Examples (Cont’d) Example . Find dim N ( A ) if A = 1 2 1 2 1 2 1 1 1 1 1 . Ans . In this case, N ( A ) = { x ∈ R 5 : A x = } The solutions of A x = are x = α u + β v , where u = (- 1 1 ) T and v = (- 1 1 1 ) T . ∴ N ( A ) = Span ( u , v ) . Note: u , v are linearly independent. Exercise Chapter 3. Vector Spaces Math1111 Basis and Dimension Examples (Cont’d) Example . Find dim N ( A ) if A = 1 2 1 2 1 2 1 1 1 1 1 ....
View Full Document

Page1 / 36

Lect16 - Chapter 3. Vector Spaces Math1111 Basis and...

This preview shows document pages 1 - 10. Sign up to view the full document.

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