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We consider an interesting consequence of the closed graph the- orem. A linear map P from a linear space X to itself is called a projection if P² = P. If P is a projection, then so is I - P and R(P) = Z(I – P), Z(P) = R(I-P). It follows that X = R(P) +Z(P) and R(P) Z(P) = {0} for every projection P defined on X. Conversely, if Y and Z are subspaces of X such that X = Y + Z and YnZ = {0}, then for every x € X there are unique y € Y and z € Z such that x = y + z, so that the linear map given by P(x) = y is a projection. It is called the projection onto Y along Z. Request clarity underlined part what is Z? Is I the identity map?

We consider an interesting consequence of the closed graph the- orem. A linear map P from a linear space X to itself is called a projection if P² = P. If P is a projection, then so is I - P and R(P) = Z(I – P), Z(P) = R(I-P). It follows that X = R(P) +Z(P) and R(P) Z(P) = {0} for every projection P defined on X. Conversely, if Y and Z are subspaces of X such that X = Y + Z and YnZ = {0}, then for every x € X there are unique y € Y and z € Z such that x = y + z, so that the linear map given by P(x) = y is a projection. It is called the projection onto Y along Z. Request clarity underlined part what is Z? Is I the identity map?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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We consider an interesting consequence of the closed graph the- orem. A linear map P from a linear space X to itself is called a projection if P² = P. If P is a projection, then so is I - P and R(P) = Z(I – P), Z(P) = R(I – P). It follows that X = R(P) + Z(P) and R(P) Z(P) = {0} for every projection P defined on X. Conversely, if Y and Z are subspaces of X such that X = Y+Z and YnZ = {0}, then for every € X there are unique y € Y and z € Z such that x = y + z, so that the linear map given by P(x) = y is a projection. It is called the projection onto Y along Z. Request clarity underlined Rail what is Z? Is I the identity map?
Transcribed Image Text:We consider an interesting consequence of the closed graph the- orem. A linear map P from a linear space X to itself is called a projection if P² = P. If P is a projection, then so is I - P and R(P) = Z(I – P), Z(P) = R(I – P). It follows that X = R(P) + Z(P) and R(P) Z(P) = {0} for every projection P defined on X. Conversely, if Y and Z are subspaces of X such that X = Y+Z and YnZ = {0}, then for every € X there are unique y € Y and z € Z such that x = y + z, so that the linear map given by P(x) = y is a projection. It is called the projection onto Y along Z. Request clarity underlined Rail what is Z? Is I the identity map?
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