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The orthogonal projection $P_{W} v$ is the $W$ -component in $V = W \oplus W^{⊥}$

Definition

Let $W$ be a subspace of a finite-dimensional inner product space $V$ . The orthogonal-decomposition theorem (For a subspace $W$ of a finite-dimensional inner product space, $V = W \oplus W^{⊥}$ ) gives

$V = W \oplus W^{⊥}$

and associates to every $v \in V$ unique vectors $w \in W$ and $z \in W^{⊥}$ with $v = w + z$ . The orthogonal projection onto $W$ is the function

$P_{W} : V ⟶ W, P_{W} v = w .$

Equivalently, $P_{W} v$ is the unique vector of $W$ such that $v - P_{W} v \in W^{⊥}$ .

Definition references

For a subspace $W$ of a finite-dimensional inner product space, $V = W \oplus W^{⊥}$

Depends on

For a subspace $W$ of a finite-dimensional inner product space, $V = W \oplus W^{⊥}$

The orthogonal projection $P_{W} v$ is the $W$ -component in $V = W \oplus W^{⊥}$

Definition

Definition references

Depends on

Dependency tree · 2 levels

Sources

The orthogonal projection PWvPW​v is the WW-component in V=W⊕W⊥V=W⊕W⊥

Definition

Definition references

Depends on

Dependency tree · 2 levels

Sources

The orthogonal projection $P_{W} v$ is the $W$ -component in $V = W \oplus W^{⊥}$