Projections and Reflections in Vector Space

(1)

http://dx.doi.org/10.4236/apm.2016.66030

Projections and Reflections in Vector Space

Kung-Kuen Tse

Department of Mathematics, Kean University, Union, NJ, USA

Received 4 April 2016; accepted 16 May 2016; published 19 May 2016

This work is licensed under the Creative Commons Attribution International License (CC BY).

http://creativecommons.org/licenses/by/4.0/

Abstract

We study projections onto a subspace and reflections with respect to a subspace in an arbitrary vector space with an inner product. We give necessary and sufficient conditions for two such transformations to commute. We then generalize the result to affine subspaces and transforma-tions.

Keywords

Projection, Reflection, Commute, Inner Product, Affine Subspace

1. Introduction

Two lines 1 and 2 in

2

 are considered. When is the reflection over 1 followed by the reflection over 2

 the same as the reflection over 2 followed by the reflection over 1? It is easy to see that it is the case if

and only if 1⊥2 or 1=2.

When considering subspaces of 3, we can ask similar questions for lines, for planes or for the mixed case of one line and one plane. Instead of addressing those cases one by one, we generalize the situation of arbitrary two linear subspaces of a vector space with an inner product.

2. Projection

Supposing that U is a vector space equipped with an inner product, V ⊂U is a linear subspace of U. Given a vector u∈U, we know from linear algebra [1] [2] that u can be decomposed uniquely as u= pV

( )

u +u′

where pV

( )

u ∈V is the projection of the vector u onto V and u′ ⊥V, i.e. U V V

⊥

= ⊕ .

Here are some elementary properties of the projection pV: 1) pV is linear.

2) u∈V if and only if pV

( )

u =u.

3) u∈V⊥ if and only if pV

( )

u =0.

( )

(2)

5) If V1 and V2 are subspaces of U, then pV₁⊕V₂

( )

u = pV₁

( )

u +pV₂

( )

u , for all u∈U . 6) If V1, V2 and W are subspaces of U, then pV₁⊕V₂

( )

W ⊂pV₁

( )

W ⊕pV₂

( )

W . 7) If V1, V2 and W are subspaces of U, then pW

(

V1⊕V2

)

= pW

( )

V1 ⊕pW

( )

V2 .

Lemma 2.1. Supposing that U is a linear space and V, W are two linear subspaces of U, if pW

( )

V = pV

( )

W then pW

( )

V =pV

( )

W =VW.

Proof. We first show that pV

( )

W =VW . Since pV

( )

W ⊂V and pV

( )

W =pW

( )

V ⊂W , we have

( )

V

p W ⊂V W . On the other hand, if u∈VW , then u∈V , hence u= pV

( )

u ∈pV

( )

W and thus

( )

V

V W⊂ p W . As a result, pV

( )

W =VW . The proof of pW

( )

V =VW is similar. 

Suppose U is a vector space and V, W are two subspaces of U. Intersecting the identity U=

(

VW

)

⊕

(

VW

)

⊥ with V and W, we get V =

(

VW

)

⊕

(

V

(

VW

)

⊥

)

and W =

(

VW

)

⊕

(

W 

(

VW

)

⊥

)

. It is obvious that these two sums are orthogonal.

Denote V′ =V

(

VW

)

⊥ and W′ =W

(

VW

)

⊥ . Using these notations, V =

(

V W

)

⊕V′ and

(

)

W = VW ⊕W′.

Lemma 2.2. pV′

( )

W′ = pW′

( )

V′ =0 if and only if pW

( )

V = pV

( )

W .

Poorf.

( )

(

)

(

)

(

)

( )

(

) { } { }

( )

(

)

( )

0 0

V V W V V W V

V W V V W V

V

p W p W p V W W

p V W p V W p W p W

V W p W

′ ′

⊕ ⊕

′ ′

′

= = ⊕

′ ′

⊂ ⊕ ⊕ ⊕

′

= ⊕ ⊕ ⊕

′

= ⊕

 



 

(⇒) If pV′

( )

W′ =0, then pV

( )

W ⊂VW. On the other hand, by the fourth property of projection above,

( )

V

V W⊂ p W . Similarly, pW

( )

V =VW . Thus, pV

( )

W = pW

( )

V .

(⇐) By Lemma 2.1, pV

( )

W =VW . For w′∈W′,

( )

V V W V V

p w′ = p _ w′ +p _′ w′ =p _′ w′

( )

V V

p w′ ∈p W =VW and p_V′

( )

w′ ∈V′, but

(

VW

)

V′ =0 , we must have pV′

( )

w′ =0 , i.e.

( )

0

V

p ′ W′ = . Similarly, pW′

( )

V′ =0. 

Theorem 2.3. Supposing that U is a vector space and V, W are two subspaces of U, then pVpW = pW pV

if and only if p_W

( )

V = p_V

( )

W .

Proof. (⇒) Assume that p_W

(

p_V

( )

u

)

= p_V

(

p_W

( )

u

)

for allu∈U. In particular,

( )

(

( )

)

(

( )

)

( )

for all .

W W V V W V

p v = p p v = p p v ∈p W v∈V

Thus, p_W

( )

V ⊂ p_V

( )

W . Similarly, p_V

( )

W ⊂p_W

( )

V .

(⇐) Assume p_W

( )

V = p_V

( )

W . By Lemma 2.2, p_V′

( )

W′ = p_W′

( )

V′ =0.

( )

(

)

(

( )

)

( )

(

)

(

( )

)

( )

(

)

(

( )

)

(

( )

)

( )

.

W V W V W V

W V W W V

W V W V W V W V

V W

p p u p p u p u

p p u p p u

p p u p p u p p u

p u

′

′ ′ ′

= +

= + +

=



 



Similarly, pV

(

pW

( )

u

)

= pV W

( )

u . 

3. Reflection over a Subspace

Supposing that U is a vector space equipped with an inner product, V ⊂U is a subspace of U. We define the refection of u∈U with respect to V as

( )

2

( )

.

V V

r u = p u −u

The above formula can be easily derived from the observation that

( )

1

(

( )

)

2

V V

p u = u+r u . Note that if u V∈ ,

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Lemma 3.1. Supposing that U is a vector space and V, W are two vector subspaces of U, then

V W W V

r r =r r if and only if pV pW = pW pV . Proof.

( )

(

)

(

( )

)

( )

(

( )

)

(

( )

)

( )

(

)

( )

2 2 2 2

4 2 2 .

W V W V V W V V

W V W V

r r u p r u r u p p u u p u u

p p u p u p u u

= − = − − −

= − − +

Similarly, r_V

(

r_W

( )

u

)

=4p_V

(

p_W

( )

u

)

−2p_V

( )

u −2p_W

( )

u +u. Hence,

if and only if .

W V V W W V V W

r r =r r p p = p p

Theorem 3.2. Supposing that U is a vector space and V, W are two subspaces of U, then rVrW =rWrV if

and only if p_V

( )

W = p_W

( )

V .

Poor. By Lemma 3.1, rV rW if and only if pV pW = pW pV. By Theorem 2.3, pV pW =pW pV if and only if p_V

( )

W =p_W

( )

V . 

4. Projection onto a Translated Subspace

Define the projection of u∈U onto a translated subspace Vˆ= +V v0 as

( )

(

)

( )

ˆ V 0 0 V V( )0 0.

V

p u = p u−v +v = p u −p v +v

ˆ V

p is well defined: supposing V+ = +v₀ V v₀′, then v₀− ∈v₀′ V. Hence p_V

( )

v₀ −p_V

( )

v′₀ = p_V

(

v₀−v₀′

)

=v₀−v′₀

and thus

( )

0 0

( )

0 0.

V V V V

p u −p v +v = p u − p v′ +v′

Theorem 4.1. p_Vˆp_Wˆ =p_Wˆ p_Vˆ if and only if VˆWˆ ≠φ and pV

( )

W = pW

( )

V .

Proof.

( )

(

)

(

( )

)

( )

(

)

( )

(

)

(

( )

)

( )

ˆ ˆ ˆ 0 0

0 0 0 0

0 0 0 0.

V V

V W W

V W W V

V W V W V V

p p u p p u p v v

p p u p w w p v v

p p u p p w p w p v v

= − +

= − + − +

Similarly, pWˆ

(

pVˆ

( )

u

)

= pW

(

pV

( )

u

)

−pW

(

pV

( )

v0

)

+pW

( )

v0 −pW

( )

w0 +w0.

Thus, pVˆ

(

pWˆ

( )

u

)

= pWˆ

(

pVˆ

( )

u

)

if and only if

( )

(

)

(

( )

)

( )

(

0

)

( )

0

( )

0 0

(

( )

0

)

( )

0

( )

0 0

V W W V

V W V V W V W W

p p u p p u

p p w p w p v v p p v p v p w w

 =

 

− + − + = − + − +



(⇒) By Theorem 2.3, the first equation implies pV

( )

W = pW

( )

V . The second equation simply means that

ˆ ˆ

V W ≠φ.

(⇐) By Theorem 2.3, the first equation is satisifed. To show the second equation, since VˆWˆ ≠φ, we have

0 0

v+v = +w w , for some v∈V and w W∈ , or v0= +w w0−v:

( )

(

)

( )

(

)

(

)

(

)

(

)

( )

(

( )

)

( )

(

)

( )

0 0 0 0

W V W V

V W V W W W V V

V W W V

p p v p v p v v

p p w w v p w w v p w w v w w v

p w p p w p v w p w p v p w p w v w w v

p p w p w p w w

− + + −

= − + − + + − + + − − − +

= − − + + + − + + − − − +

= − + + −

       

which is the second equation.

5. Reflection over a Translated Subspace

We next discuss the reflection over a translated subspace. Let V⊂U be a subspace. A translated subspace is

(4)

( )

(

)

( )

ˆ V 0 0 V V 0 0.

V

r u =r u−v +v =r u −r v +v

ˆ V

r is well-defined: supposing V+ = +v0 V v0′, then v0− ∈v0′ V and hence r vV

( )

0 −r vV

( )

0′ =r vV

(

0−v0′

)

=v0−v′0.

As a result,

( )

0 0 ( )

( )

0 0.

V V V V

r u −r v +v =r u −r v′ +v′

Supposing Wˆ =W +w₀ for some w0∈W is another translated subspace.

( )

(

)

(

( )

)

( )

(

)

( )

(

)

(

( )

)

( )

ˆ ˆ ˆ 0 0

0 0 0 0

0 0 0 0.

W W

W V V

W V V W

W V W V W W

r r u r r u r w w

r r u r v v r w w

r r u r r v r v r w w

= − +

= − + − +

Similarly, rVˆ

(

rWˆ

( )

u

)

=rV

(

rW

( )

u

)

−rV

(

rW

( )

w0

)

+rV

( )

w0 −rV

( )

v0 +v0.

Theorem 5.1. r_Wˆ r_Vˆ =r_Vˆr_Wˆ if and only if VˆWˆ ≠φ and pV

( )

W = pW

( )

V .

Proof. rWˆ

(

rVˆ

( )

u

)

=rVˆ

(

rWˆ

( )

u

)

if and only if

( )

(

)

(

( )

)

( )

(

0

)

( )

0

( )

0 0

(

( )

0

)

( )

0

( )

0 0

W V V W

W V W W V W V V

r r u r r u

p p v p v p w w p p w p w p v v

 =

 

− + − + = − + − +



(⇒) By Theorem 3.2, rVrW =rWrV implies pV

( )

W = pW

( )

V . The second equation simply means

ˆ ˆ

V W ≠φ.

(⇐) We express rWˆ

(

rVˆ

( )

u

)

and rVˆ

(

rWˆ

( )

u

)

in terms of projections:

( )

(

)

(

( )

)

(

( )

)

( )

ˆ ˆ W V 4 W V 0 4 W 0 2 V 0 2 0 2 W 0 2 0,

W V

r r u =r r u − p p v + p v + p v − v − p w + w

( )

(

)

(

( )

)

(

( )

)

( )

ˆ ˆ V W 4 V W 0 4 V 0 2 W 0 2 0 2 V 0 2 .0

V W

r r u =r r u − p p w + p w + p w − w − p v + v

By Theorem 3.2, p W_V

( )

= p_W

( )

V implies rVrW =rWrV . By Lemma 3.1, we also have pVpW =pWpV. To show rWˆ rVˆ =rVˆrWˆ, it suffices to verify the second equation

( )

(

0

)

( )

0

( )

0 0

(

( )

0

)

( )

0

( )

0 0.

W V W V V W V W

p p v p v p v v p p w p w p w w

− + + − = − + + −

Since VˆWˆ ≠φ, we must have v+v0= +w w0 for some v∈V and w W∈ , or v0= +w w0−v:

( )

(

)

( )

(

)

(

)

(

)

(

)

( )

(

)

(

( )

)

( )

(

)

(

( )

)

( )

(

)

(

( )

)

( )

(

( )

)

0 0 0 0

0 0

0 0 0 0

0

W V W V

W V W V W W W

V V

W V W V W V V

V W V W W V V

V V W

p p v p v p v v

p p w w v p w w v p w w v w w v

p p w p p w p v w p w p v

p w p w v w w v

p p w p p w p w p w p w w

p w p p w

− + + − = − + − + + − + + − − − + = − − + + + − + + − − − + = − − + + + − = − − + + + − = − − +                     

( )

(

)

( )

0 0 0

0 0 0 0

W V V

V W V W

p w p w p w w

p p w p w p w w

+ + −

= − + + −



6. Mixed Transformations

Theorem 6.1. p_Vˆr_Wˆ =r_Wˆ p_Vˆ if and only if VˆWˆ ≠φ and pV

( )

W = pW

( )

V .

Theorem 6.2. p_Vˆr_Wˆ =r_Vˆp_Wˆ if and only if VˆWˆ ≠φ and V =W.

Theorem 6.3. p_Vˆr_Wˆ = p_Wˆ r_Vˆ if and only if VˆWˆ ≠φ and V =W.

7. Generalizations

(5)

Theorem 7.1.

( )1 ( )n ( )1 ( )n ,

V V V V n

p_σ p_σ = p_τ p_τ for allσ τ∈Σ

if and only if

( )1 (n1)

( )

( ) ( )1 (n1)

( )

( ) , .

V V n V V n n

p p V p p V for all

σ  σ − σ = τ  τ − τ σ τ∈Σ

Theorem 7.2.

( )1 ( )n ( )1 ( )n ,

V V V V n

r_σ r_σ =r_τ r_τ for allσ τ∈Σ

if and only if

( )1 (n1)

( )

( ) ( )1 (n1)

( )

( ) , .

V V n V V n n

p_σ p_σ ₋ V_σ = p_τ p_τ ₋ V_τ for allσ τ∈Σ

Theorem 7.3.

( )1 ( ) ( )1 ( )

ˆ ˆ ˆ ˆ ,

n n n

V V V V

p p p p for all

σ  σ = τ  τ σ τ∈Σ if and only if VˆiVˆj≠φ for all i j, and

( )1 (n1)

( )

( ) ( )1 (n1)

( )

( ) , .

V V n V V n n

σ  σ − σ = τ  τ − τ σ τ∈Σ

Theorem 7.4.

( )1 ( ) ( )1 ( )

ˆ ˆ ˆ ˆ ,

n n n

V V V V

r r r r for all

σ  σ = τ  τ σ τ∈Σ if and only if VˆiVˆj≠φ for all i j, and

( )1 (n1)

( )

( ) ( )1 (n1)

( )

( ) , .

V V n V V n n

σ  σ − σ = τ  τ − τ σ τ∈Σ

References