Lect12 - Chapter 3. Vector Spaces Math1111 Subspaces...

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Unformatted text preview: Chapter 3. Vector Spaces Math1111 Subspaces Example (Cont’d) Example . (a) Let T = { ( x y ) T : x + 2 y = } . Describe T geometrically. Is T a subspace of R 2 ? (b) Let S = { ( x y ) T : x + 2 y = 1 } . Describe S geometrically. Is S a subspace of R 2 ? (c) Let ` : ax + by = c be a straight line on R 2 . Let W be the set of all vectors in R 2 that represent points on ` . Is W a subspace of R 2 ? Chapter 3. Vector Spaces Math1111 Nullspace of a Matrix Definition Let A be an m × n matrix, and N ( A ) = x ∈ R n : A x = . Claim: N ( A ) is a subspace of R n . Chapter 3. Vector Spaces Math1111 Nullspace of a Matrix Definition Let A be an m × n matrix, and N ( A ) = x ∈ R n : A x = . Claim: N ( A ) is a subspace of R n . Definition We call N ( A ) the nullspace of A . Chapter 3. Vector Spaces Math1111 Nullspace of a Matrix Definition Let A be an m × n matrix, and N ( A ) = x ∈ R n : A x = . Claim: N ( A ) is a subspace of R n . Definition We call N ( A ) the nullspace of A . Example . Determine N ( A ) if A = 1 1 1 2 1 1 . Chapter 3. Vector Spaces Math1111 Span of Vectors Motivation Recall in general a nonempty subset of a vector space V is not a subspace. e.g Let 6 = x ∈ V . Then S = { x } is not a vector space. Chapter 3. Vector SpacesChapter 3....
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This note was uploaded on 09/06/2010 for the course MATH MATH1111 taught by Professor Forgot during the Fall '08 term at HKU.

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Lect12 - Chapter 3. Vector Spaces Math1111 Subspaces...

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