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

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Unformatted text preview: Chapter 3. Vector Spaces Math1111 Subspaces Definition & Examples Definition Let ; 6 = 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 . Chapter 3. Vector Spaces Math1111 Subspaces Definition & Examples Definition Let ; 6 = 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 R 3 . Chapter 3. Vector Spaces Math1111 Subspaces Definition & Examples Definition Let ; 6 = 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 R 3 . Example . Show that S = ( x 1 ) T : x ∈ R is a not 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 = ....
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Lect13 - Chapter 3. Vector Spaces Math1111 Subspaces...

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