Lect12

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

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Chapter 3. Vector Spaces Math1111 Subspaces Example (Cont’d) Example . (a) Let T = { ( x y ) T : x + 2 y = 0 } . Describe T geometrically. Is T a subspace of 2 ? (b) Let S = { ( x y ) T : x + 2 y = 1 } . Describe S geometrically. Is S a subspace of 2 ? (c) Let : ax + by = c be a straight line on 2 . Let W be the set of all vectors in 2 that represent points on . Is W a subspace of 2 ?

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Chapter 3. Vector Spaces Math1111 Nullspace of a Matrix Definition Let A be an m × n matrix, and N ( A ) = x n : A x = 0 . Claim: N ( A ) is a subspace of n .
Chapter 3. Vector Spaces Math1111 Nullspace of a Matrix Definition Let A be an m × n matrix, and N ( A ) = x n : A x = 0 . Claim: N ( A ) is a subspace of n . Definition We call N ( A ) the nullspace of A .

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Chapter 3. Vector Spaces Math1111 Nullspace of a Matrix Definition Let A be an m × n matrix, and N ( A ) = x n : A x = 0 . Claim: N ( A ) is a subspace of n . Definition We call N ( A ) the nullspace of A . Example . Determine N ( A ) if A = 1 1 1 0 2 1 0 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 0 = x V . Then S = { x } is not a vector space.

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Chapter 3. Vector Spaces
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