Orthonormal Bases; Gram-Schmidt Process;
QR-Decomposition
MATH 322, Linear Algebra I
J. Robert Buchanan
Department of Mathematics
Spring 2007
J. Robert Buchanan Orthonormal Bases; Gram-Schmidt Pro cess; QR-Decomposition
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Orthonormal Bases, Gram-Schmidt Process, and QR-Decomposition in Linear Algebra, Assignments of Linear Algebra
This document from math 322, linear algebra i, covers the concepts of orthonormal bases, gram-schmidt process, and qr-decomposition in the context of inner product spaces. The motivation for using orthonormal bases, the definitions of orthogonal sets and orthonormal bases, and provides examples and theorems on coordinate representations, linear independence, and orthogonal projections. The document also discusses the gram-schmidt process and its application to qr-decomposition.
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Orthonormal Bases; Gram-Schmidt Process;
QR -Decomposition
MATH 322, Linear Algebra I
J. Robert Buchanan
Department of Mathematics
Spring 2007
Motivation
When working with an inner product space, the most convenient bases are those that (^1) consist of orthogonal vectors, and (^2) consist of vectors of length 1.
Orthogonal Sets
Definition A set of vectors in an inner product space is an orthogonal set if all the vectors are pairwise orthogonal. An orthogonal set in which each vector has norm (length) 1 is called an orthonormal set.
Example The standard basis { e 1 , e 2 ,... , e n } is an orthonormal set in R n.
Recall: if v ∈ V and v 6 = 0 then
u =
v ‖ v ‖
has norm 1.
Orthogonal Sets
Definition A set of vectors in an inner product space is an orthogonal set if all the vectors are pairwise orthogonal. An orthogonal set in which each vector has norm (length) 1 is called an orthonormal set.
Example The standard basis { e 1 , e 2 ,... , e n } is an orthonormal set in R n.
Recall: if v ∈ V and v 6 = 0 then
u =
v ‖ v ‖
has norm 1.
Orthonormal Basis
Definition A basis for an inner product space consisting of orthonormal vectors is called an orthonormal basis. A basis for an inner product space consisting of orthogonal vectors is called an orthogonal basis.
Theorem If B = { v 1 , v 2 ,... , v n } is an orthonormal basis for an inner product space V , and if u ∈ V , then
u = 〈 u , v 1 〉 v 1 + 〈 u , v 2 〉 v 2 + · · · + 〈 u , v n 〉 v n.
Remark: The coordinates of u relative to B are 〈 u , v 1 〉, 〈 u , v 2 〉,
... , 〈 u , v n 〉. Proof.
Orthonormal Basis
Definition A basis for an inner product space consisting of orthonormal vectors is called an orthonormal basis. A basis for an inner product space consisting of orthogonal vectors is called an orthogonal basis.
Theorem If B = { v 1 , v 2 ,... , v n } is an orthonormal basis for an inner product space V , and if u ∈ V , then
u = 〈 u , v 1 〉 v 1 + 〈 u , v 2 〉 v 2 + · · · + 〈 u , v n 〉 v n.
Remark: The coordinates of u relative to B are 〈 u , v 1 〉, 〈 u , v 2 〉,
... , 〈 u , v n 〉. Proof.
Coordinates Relative to Orthonormal Bases
Theorem If B is an orthonormal basis for an n-dimensional inner product space, and if
( u )B = ( u 1 , u 2 ,... , un ) and ( v )B = ( v 1 , v 2 ,... , vn )
then (^1) ‖ u ‖ =
u 12 + u 22 + · · · u n^2
(^2) d ( u , v ) =
( u 1 − v 1 )^2 + ( u 2 − v 2 )^2 + · · · + ( un − vn )^2 (^3) 〈 u , v 〉 = u 1 v 1 + u 2 v 2 + · · · + unvn
Remark: computing norms, distances, and inner products with coordinates relative to orthonormal bases is equivalent to computing them in Euclidean coordinates.
Coordinates Relative to Orthonormal Bases
Theorem If B is an orthonormal basis for an n-dimensional inner product space, and if
( u )B = ( u 1 , u 2 ,... , un ) and ( v )B = ( v 1 , v 2 ,... , vn )
then (^1) ‖ u ‖ =
u 12 + u 22 + · · · u n^2
(^2) d ( u , v ) =
( u 1 − v 1 )^2 + ( u 2 − v 2 )^2 + · · · + ( un − vn )^2 (^3) 〈 u , v 〉 = u 1 v 1 + u 2 v 2 + · · · + unvn
Remark: computing norms, distances, and inner products with coordinates relative to orthonormal bases is equivalent to computing them in Euclidean coordinates.
Linear Independence
Theorem If B = { v 1 , v 2 ,... , v n } is an orthogonal set of nonzero vectors in an inner product space, then B is linearly independent.
Proof.
Linear Independence
Theorem If B = { v 1 , v 2 ,... , v n } is an orthogonal set of nonzero vectors in an inner product space, then B is linearly independent.
Proof.
Orthogonal Projections
Question: how do you create an orthogonal basis for an arbitrary finite-dimensional inner product space?
Theorem (Projection Theorem) If W is a finite-dimensional subspace of an inner product space V , then every vector u ∈ V can expressed uniquely as
u = w 1 + w 2
where w 1 ∈ W and w 2 ∈ W ⊥.
Remark: w 1 is called the orthogonal projection of u on W. w 2 is called the component of u orthogonal to W.
Proof. Later
Orthogonal Projections
Question: how do you create an orthogonal basis for an arbitrary finite-dimensional inner product space?
Theorem (Projection Theorem) If W is a finite-dimensional subspace of an inner product space V , then every vector u ∈ V can expressed uniquely as
u = w 1 + w 2
where w 1 ∈ W and w 2 ∈ W ⊥.
Remark: w 1 is called the orthogonal projection of u on W. w 2 is called the component of u orthogonal to W.
Proof. Later
Illustration
u = w 1 + w 2 = proj W u + proj W ⊥ u Thus
proj W ⊥ u = u − proj W u u = proj W u + ( u − proj W u )
w 2
w 1
u u-projwu
projWu
u
Calculating Orthogonal Projections
Theorem Suppose W is an finite-dimensional subspace of an inner product space V. (^1) If B = { v 1 , v 2 ,... , v r } is an orthonormal basis for W and u ∈ V then
proj W u = 〈 u , v 1 〉 v 1 + 〈 u , v 2 〉 v 2 + · · · + 〈 u , v r 〉 v r.
(^2) If B′^ = { v 1 , v 2 ,... , v r } is an orthogonal basis for W and u ∈ V then
proj W u = 〈 u , v 1 〉 ‖ v 1 ‖^2
v 1 + 〈 u , v 2 〉 ‖ v 2 ‖^2
v 2 + · · · + 〈 u , v r 〉 ‖ v r ‖^2
v r.