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 1 € 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?
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 1 € 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?
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.5: The Kernel And Range Of A Linear Transformation
Problem 4EQ
Related questions
Question
100%
![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?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe15b7304-cc73-4505-92c3-23aa2fda4f71%2F3df5fff6-12d5-49a0-92c5-6f5d021c5f82%2F47gr5qq_processed.png&w=3840&q=75)
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?
Expert Solution
![](/static/compass_v2/shared-icons/check-mark.png)
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 2 images
![Blurred answer](/static/compass_v2/solution-images/blurred-answer.jpg)
Recommended textbooks for you
![Linear Algebra: A Modern Introduction](https://www.bartleby.com/isbn_cover_images/9781285463247/9781285463247_smallCoverImage.gif)
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
![Elementary Linear Algebra (MindTap Course List)](https://www.bartleby.com/isbn_cover_images/9781305658004/9781305658004_smallCoverImage.gif)
Elementary Linear Algebra (MindTap Course List)
Algebra
ISBN:
9781305658004
Author:
Ron Larson
Publisher:
Cengage Learning
![Linear Algebra: A Modern Introduction](https://www.bartleby.com/isbn_cover_images/9781285463247/9781285463247_smallCoverImage.gif)
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
![Elementary Linear Algebra (MindTap Course List)](https://www.bartleby.com/isbn_cover_images/9781305658004/9781305658004_smallCoverImage.gif)
Elementary Linear Algebra (MindTap Course List)
Algebra
ISBN:
9781305658004
Author:
Ron Larson
Publisher:
Cengage Learning