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Lect13

# Lect13 - Chapter 3 Vector Spaces Math1111 Subspaces...

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Chapter 3. Vector Spaces Math1111 Subspaces Definition & Examples Definition Let = S V where V is a vector space. If (i) α x S for any scalar α and any x S , and (ii) x + y S whenever x , y S , then S is said to be a subspace of V .

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Chapter 3. Vector Spaces Math1111 Subspaces Definition & Examples Definition Let = S V where V is a vector space. If (i) α x S for any scalar α and any x S , and (ii) x + y S whenever x , y S , then S is said to be a subspace of V . Example . Show that S = ( x 1 x 2 x 3 ) T : x 1 = x 2 is a subspace of 3 .
Chapter 3. Vector Spaces Math1111 Subspaces Definition & Examples Definition Let = S V where V is a vector space. If (i) α x S for any scalar α and any x S , and (ii) x + y S whenever x , y S , then S is said to be a subspace of V . Example . Show that S = ( x 1 x 2 x 3 ) T : x 1 = x 2 is a subspace of 3 . Example . Show that S = ( x 1 ) T : x is a not 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 .
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