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Subspace

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Definition

LinearAlgebra

Definition

A subspace of a vector space $V$ is a subset $U\subset V$ that, together with the addition and scalar multiplication on $V$ , is itself a vector space. ## Remark If $U_1,U_2\subset V$ are subspaces, then $0\in U_1$ and $0\in U_2$ . If $u_1\in U_1$ and $u_2\in U_2$ then $u_1+0=u_1\in U_1+U_2$ and the same for $u_2\in U_2$ . Therefore, the union of $U_1$ and $U_2$ is also contained in $U_1+U_2$ . In fact, $U_1+U_2$ is the smallest subspace containing $U_1\cup U_2$

Linked from

Eigenvector

Orthogonal Complement

Complementary Subspaces

Direct Sum

Invariant Subspace

Invariant

Subspace

Kernel and Image are Subspaces

Conditions for Diagonalizability

Complement of Subspace Generates Direct Sum

Double Complement Returns the Original Subspace

Properties of Orthogonal Complements

The Orthogonal Complement is a Complementary Subspace

Criterion for Subspace

Span of Vectors is a Subspace of the Vector Space

Sum of Subspaces is Smallest Subspace Containing their Union

Sum of Subspaces is a Subspace

Controllability Normal Form