Orthogonal Complements Notation

F denotes either R or C.

V denotes an inner product space over F. If U is a subset of V, then U⊥ is a subspace of V. {0}⊥ = V. V⊥ = {0}. If U is a subset of V, then If U is a line in R3 containing the origin, U ∩ U⊥ ⊂ {0}. ⊥ then U is the plane containing the If U and W are subsets of V origin that is perpendicular to U. and U ⊂ W, then W⊥ ⊂ U⊥. If U is a plane in R3 containing the origin, then U⊥ is the line containing the origin that is perpendicular to U.

Basic properties of orthogonal complement

Orthogonal Complements

Deﬁnition: orthogonal complement, U⊥

If U is a subset of V, then the orthogonal complement of U, denoted U⊥, is the set of all vectors in V that are orthogonal to every vector in U: U⊥ = {v ∈ V : hv, ui = 0 for every u ∈ U}. If U is a subset of V, then U⊥ is a subspace of V. {0}⊥ = V. V⊥ = {0}. If U is a subset of V, then U ∩ U⊥ ⊂ {0}. If U and W are subsets of V and U ⊂ W, then W⊥ ⊂ U⊥.

Basic properties of orthogonal complement

If U is a plane in R3 containing the origin, then U⊥ is the line containing the origin that is perpendicular to U.

Orthogonal Complements

Deﬁnition: orthogonal complement, U⊥

If U is a subset of V, then the orthogonal complement of U, denoted U⊥, is the set of all vectors in V that are orthogonal to every vector in U: U⊥ = {v ∈ V : hv, ui = 0 for every u ∈ U}.

If U is a line in R3 containing the origin, then U⊥ is the plane containing the origin that is perpendicular to U. If U is a subset of V, then U⊥ is a subspace of V. {0}⊥ = V. V⊥ = {0}. If U is a subset of V, then U ∩ U⊥ ⊂ {0}. If U and W are subsets of V and U ⊂ W, then W⊥ ⊂ U⊥.

Basic properties of orthogonal complement

Orthogonal Complements

Deﬁnition: orthogonal complement, U⊥

If U is a subset of V, then the orthogonal complement of U, denoted U⊥, is the set of all vectors in V that are orthogonal to every vector in U: U⊥ = {v ∈ V : hv, ui = 0 for every u ∈ U}.

If U is a line in R3 containing the origin, then U⊥ is the plane containing the origin that is perpendicular to U. If U is a plane in R3 containing the origin, then U⊥ is the line containing the origin that is perpendicular to U. If U is a subset of V, then U⊥ is a subspace of V. {0}⊥ = V. V⊥ = {0}. If U is a subset of V, then U ∩ U⊥ ⊂ {0}. If U and W are subsets of V and U ⊂ W, then W⊥ ⊂ U⊥.

Basic properties of orthogonal complement

Orthogonal Complements

Deﬁnition: orthogonal complement, U⊥

If U is a subset of V, then the orthogonal complement of U, denoted U⊥, is the set of all vectors in V that are orthogonal to every vector in U: U⊥ = {v ∈ V : hv, ui = 0 for every u ∈ U}.

If U is a line in R3 containing the origin, then U⊥ is the plane containing the origin that is perpendicular to U. If U is a plane in R3 containing the origin, then U⊥ is the line containing the origin that is perpendicular to U. {0}⊥ = V. V⊥ = {0}. If U is a subset of V, then U ∩ U⊥ ⊂ {0}. If U and W are subsets of V and U ⊂ W, then W⊥ ⊂ U⊥.

Orthogonal Complements

If U is a line in R3 containing the origin, then U⊥ is the plane containing the origin that is perpendicular to U. If U is a plane in R3 containing the origin, then U⊥ is the line containing the origin that is perpendicular to U. V⊥ = {0}. If U is a subset of V, then U ∩ U⊥ ⊂ {0}. If U and W are subsets of V and U ⊂ W, then W⊥ ⊂ U⊥.

Orthogonal Complements

If U is a line in R3 containing the origin, then U⊥ is the plane containing the origin that is perpendicular to U. If U is a plane in R3 containing the origin, then U⊥ is the line containing the origin that is perpendicular to U. If U is a subset of V, then U ∩ U⊥ ⊂ {0}. If U and W are subsets of V and U ⊂ W, then W⊥ ⊂ U⊥.

Orthogonal Complements

If U is a line in R3 containing the origin, then U⊥ is the plane containing the origin that is perpendicular to U. If U is a plane in R3 containing the origin, then U⊥ is the line containing the origin that is perpendicular to U. If U and W are subsets of V and U ⊂ W, then W⊥ ⊂ U⊥.

Orthogonal Complements

Deﬁnition: orthogonal complement, U⊥ Basic properties of orthogonal complement If U is a subset of V, then the orthogonal ⊥ complement of U, denoted U , is the set of If U is a subset of V, then U⊥ is all vectors in V that are orthogonal to every a subspace of V. vector in U: {0}⊥ = V. ⊥ U = {v ∈ V : hv, ui = 0 for every u ∈ U}. V⊥ = {0}. If U is a subset of V, then If U is a line in R3 containing the origin, U ∩ U⊥ ⊂ {0}. then U⊥ is the plane containing the origin that is perpendicular to U. If U is a plane in R3 containing the origin, then U⊥ is the line containing the origin that is perpendicular to U. Orthogonal Complements

Deﬁnition: orthogonal complement, U⊥ Basic properties of orthogonal complement If U is a subset of V, then the orthogonal ⊥ complement of U, denoted U , is the set of If U is a subset of V, then U⊥ is all vectors in V that are orthogonal to every a subspace of V. vector in U: {0}⊥ = V. ⊥ U = {v ∈ V : hv, ui = 0 for every u ∈ U}. V⊥ = {0}. If U is a subset of V, then If U is a line in R3 containing the origin, U ∩ U⊥ ⊂ {0}. ⊥ then U is the plane containing the If U and W are subsets of V origin that is perpendicular to U. and U ⊂ W, then W⊥ ⊂ U⊥. If U is a plane in R3 containing the origin, then U⊥ is the line containing the origin that is perpendicular to U. Dimension of the orthogonal complement

Suppose V is ﬁnite-dimensional and U is a subspace of V. Then dim U⊥ = dim V − dim U.

Proof Suppose v ∈ V. Let e1,..., em be an orthonormal basis of U. Let The orthogonal complement of the orthogonal complement u = hv, e1ie1 + ··· + hv, emiem.

Then hv − u, eji = hv, eji − hu, eji = 0 for Suppose U is a ﬁnite-dimensional sub- ⊥ j = 1,..., m. Thus v − u ∈ U . Now space of V. Then v = u + (v − u), U = (U⊥)⊥. showing that v ∈ U+U⊥. Thus V = U+U⊥.

Direct Sum

Direct sum of a subspace and its orthogonal complement

Suppose U is a ﬁnite-dimensional subspace of V. Then V = U ⊕ U⊥. Dimension of the orthogonal complement

Suppose V is ﬁnite-dimensional and U is a subspace of V. Then dim U⊥ = dim V − dim U.

The orthogonal complement of the orthogonal complement

Direct Sum

Direct sum of a subspace and its orthogonal complement

Suppose U is a ﬁnite-dimensional subspace of V. Then V = U ⊕ U⊥.

Proof Suppose v ∈ V. Let e1,..., em be an orthonormal basis of U. Let

u = hv, e1ie1 + ··· + hv, emiem. Dimension of the orthogonal complement

Suppose V is ﬁnite-dimensional and U is a subspace of V. Then dim U⊥ = dim V − dim U.

The orthogonal complement of the orthogonal complement

Suppose U is a ﬁnite-dimensional sub- ⊥ Thus v − u ∈ U . Now space of V. Then v = u + (v − u), U = (U⊥)⊥. showing that v ∈ U+U⊥. Thus V = U+U⊥.

Direct Sum

Direct sum of a subspace and its orthogonal complement

Suppose U is a ﬁnite-dimensional subspace of V. Then V = U ⊕ U⊥.

Proof Suppose v ∈ V. Let e1,..., em be an orthonormal basis of U. Let

u = hv, e1ie1 + ··· + hv, emiem.

Then hv − u, eji = hv, eji − hu, eji = 0 for j = 1,..., m. Dimension of the orthogonal complement

Suppose V is ﬁnite-dimensional and U is a subspace of V. Then dim U⊥ = dim V − dim U.

The orthogonal complement of the orthogonal complement

Suppose U is a ﬁnite-dimensional sub- Now space of V. Then v = u + (v − u), U = (U⊥)⊥. showing that v ∈ U+U⊥. Thus V = U+U⊥.

Direct Sum

Direct sum of a subspace and its orthogonal complement

Suppose U is a ﬁnite-dimensional subspace of V. Then V = U ⊕ U⊥.

Proof Suppose v ∈ V. Let e1,..., em be an orthonormal basis of U. Let

u = hv, e1ie1 + ··· + hv, emiem.

Then hv − u, eji = hv, eji − hu, eji = 0 for j = 1,..., m. Thus v − u ∈ U⊥. Dimension of the orthogonal complement

Suppose V is ﬁnite-dimensional and U is a subspace of V. Then dim U⊥ = dim V − dim U.

The orthogonal complement of the orthogonal complement

Suppose U is a ﬁnite-dimensional subspace of V. Then U = (U⊥)⊥.

Direct Sum

Direct sum of a subspace and its orthogonal complement

Suppose U is a ﬁnite-dimensional subspace of V. Then V = U ⊕ U⊥.

Proof Suppose v ∈ V. Let e1,..., em be an orthonormal basis of U. Let

u = hv, e1ie1 + ··· + hv, emiem.

Then hv − u, eji = hv, eji − hu, eji = 0 for j = 1,..., m. Thus v − u ∈ U⊥. Now v = u + (v − u), showing that v ∈ U+U⊥. Thus V = U+U⊥. The orthogonal complement of the orthogonal complement

Suppose U is a ﬁnite-dimensional subspace of V. Then U = (U⊥)⊥.

Direct Sum

Direct sum of a subspace and its Dimension of the orthogonal orthogonal complement complement

Suppose U is a ﬁnite-dimensional sub- Suppose V is ﬁnite-dimensional and U space of V. Then is a subspace of V. Then ⊥ V = U ⊕ U . dim U⊥ = dim V − dim U.

Proof Suppose v ∈ V. Let e1,..., em be an orthonormal basis of U. Let

u = hv, e1ie1 + ··· + hv, emiem.

Then hv − u, eji = hv, eji − hu, eji = 0 for j = 1,..., m. Thus v − u ∈ U⊥. Now v = u + (v − u), showing that v ∈ U+U⊥. Thus V = U+U⊥. Direct Sum

Direct sum of a subspace and its Dimension of the orthogonal orthogonal complement complement

Suppose U is a ﬁnite-dimensional sub- Suppose V is ﬁnite-dimensional and U space of V. Then is a subspace of V. Then ⊥ V = U ⊕ U . dim U⊥ = dim V − dim U.

Proof Suppose v ∈ V. Let e1,..., em be an orthonormal basis of U. Let The orthogonal complement of the orthogonal complement u = hv, e1ie1 + ··· + hv, emiem.

Then hv − u, eji = hv, eji − hu, eji = 0 for Suppose U is a ﬁnite-dimensional sub- ⊥ j = 1,..., m. Thus v − u ∈ U . Now space of V. Then v = u + (v − u), U = (U⊥)⊥. showing that v ∈ U+U⊥. Thus V = U+U⊥. PU ∈ L(V) ;

PUu = u for every u ∈ U; ⊥ PUw = 0 for every w ∈ U ;

range PU = U; ⊥ null PU = U ; ⊥ v − PUv ∈ U for every v ∈ V; 2 PU = PU;

kPUvk ≤ kvk for every v ∈ V.

Properties of the orthogonal projection PU

Suppose U is a ﬁnite-dimensional subspace of V. Then

If e1,..., em is an orthonormal basis of U, then

PUv = hv, e1ie1 + ··· + hv, emiem.

Orthogonal Projection

Deﬁnition: orthogonal projection, PU

Suppose U is a ﬁnite-dimensional subspace of V. The orthogonal projection of V onto U is the operator PU ∈ L(V) deﬁned as follows: For v ∈ V, write v = u + w, where u ∈ U ⊥ and w ∈ U . Then PUv = u. PU ∈ L(V) ;

PUu = u for every u ∈ U; ⊥ PUw = 0 for every w ∈ U ;

range PU = U; ⊥ null PU = U ; ⊥ v − PUv ∈ U for every v ∈ V; 2 PU = PU;

kPUvk ≤ kvk for every v ∈ V.

Properties of the orthogonal projection PU

Suppose U is a ﬁnite-dimensional subspace of V. Then

Orthogonal Projection

Deﬁnition: orthogonal projection, PU

If e1,..., em is an orthonormal basis of U, then

PUv = hv, e1ie1 + ··· + hv, emiem. PUu = u for every u ∈ U; ⊥ PUw = 0 for every w ∈ U ;

range PU = U; ⊥ null PU = U ; ⊥ v − PUv ∈ U for every v ∈ V; 2 PU = PU;

kPUvk ≤ kvk for every v ∈ V.

Orthogonal Projection

Definition: orthogonal projection, PU Properties of the orthogonal projection PU Suppose U is a finite-dimensional subspace of V. The orthogonal projection of V Suppose U is a finite-dimensional onto U is the operator PU ∈ L(V) defined subspace of V. Then as follows: PU ∈ L(V) ; For v ∈ V, write v = u + w, where u ∈ U ⊥ and w ∈ U . Then PUv = u.

If e1,..., em is an orthonormal basis of U, then

PUv = hv, e1ie1 + ··· + hv, emiem. ⊥ PUw = 0 for every w ∈ U ;

range PU = U; ⊥ null PU = U ; ⊥ v − PUv ∈ U for every v ∈ V; 2 PU = PU;

kPUvk ≤ kvk for every v ∈ V.

Orthogonal Projection

If e1,..., em is an orthonormal basis of U, then

PUv = hv, e1ie1 + ··· + hv, emiem. range PU = U; ⊥ null PU = U ; ⊥ v − PUv ∈ U for every v ∈ V; 2 PU = PU;

kPUvk ≤ kvk for every v ∈ V.

Orthogonal Projection

If e1,..., em is an orthonormal basis of U, then

PUv = hv, e1ie1 + ··· + hv, emiem. ⊥ null PU = U ; ⊥ v − PUv ∈ U for every v ∈ V; 2 PU = PU;

kPUvk ≤ kvk for every v ∈ V.

Orthogonal Projection

range PU = U; If e1,..., em is an orthonormal basis of U, then

PUv = hv, e1ie1 + ··· + hv, emiem. ⊥ v − PUv ∈ U for every v ∈ V; 2 PU = PU;

kPUvk ≤ kvk for every v ∈ V.

Orthogonal Projection

range PU = U; If e1,..., em is an orthonormal basis of U, ⊥ then null PU = U ;

PUv = hv, e1ie1 + ··· + hv, emiem. 2 PU = PU;

kPUvk ≤ kvk for every v ∈ V.

Orthogonal Projection

range PU = U; If e1,..., em is an orthonormal basis of U, ⊥ then null PU = U ; ⊥ v − PUv ∈ U for every v ∈ V; PUv = hv, e1ie1 + ··· + hv, emiem. kPUvk ≤ kvk for every v ∈ V.

Orthogonal Projection

range PU = U; If e1,..., em is an orthonormal basis of U, ⊥ then null PU = U ; ⊥ v − PUv ∈ U for every v ∈ V; PUv = hv, e1ie1 + ··· + hv, emiem. 2 PU = PU; Orthogonal Projection

range PU = U; If e1,..., em is an orthonormal basis of U, ⊥ then null PU = U ; ⊥ v − PUv ∈ U for every v ∈ V; PUv = hv, e1ie1 + ··· + hv, emiem. 2 PU = PU;

kPUvk ≤ kvk for every v ∈ V. Linear Algebra Done Right, by Sheldon Axler