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Direct Sum PDF

The document defines the direct sum of two subspaces U and V of Rn. The direct sum U ⊕ V is the sum of the subspaces U and V, where their intersection is only the zero vector. For any vector z in the direct sum U ⊕ V, there exist unique vectors u in U and v in V such that z is their sum.

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Sarit Burman
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351 views1 page

Direct Sum PDF

The document defines the direct sum of two subspaces U and V of Rn. The direct sum U ⊕ V is the sum of the subspaces U and V, where their intersection is only the zero vector. For any vector z in the direct sum U ⊕ V, there exist unique vectors u in U and v in V such that z is their sum.

Uploaded by

Sarit Burman
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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Direct sum of subspaces

Fact: Let A be an m × n matrix. Then U := {x ∈ Rn : Ax = 0} is


a subspace of Rn , called the nullspace of A.

Definition: Let U and V be two subspaces of Rn . Then

U + V := {u + v : u ∈ U, v ∈ V }

is called the sum of the subspaces U and V .

Definition: Let U and V be two subspaces of Rn . If U ∩ V = {0}


then the sum U + V is called the direct sum of U and V and is
denoted by U ⊕ V . Thus

U ⊕ V = U + V and U ∩ V = {0}.

Fact: Let U and V be subspaces of Rn . Then U + V and U ⊕ V


are subspaces of Rn . If z ∈ U ⊕ V then there exist unique u ∈ U
and v ∈ V such that z = u + v.
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