Is this representation of a linear affine space unique?
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Suppose $S subset mathbb R^n$ is a linear affine subspace. I know picking any $s in S$, $S- s =: U$ is a subspace and we can write $S = s + U$. Now consider writing $s = s_{U} + s_{U^{perp}}$ where the subscripts denote the orthogonal projection. Then $S = s_{U^{perp}} + U$. I would like to know whether this representation, i.e., $s_{U^{perp}}$, is a uniquely defined vector.
linear-algebra projection
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Suppose $S subset mathbb R^n$ is a linear affine subspace. I know picking any $s in S$, $S- s =: U$ is a subspace and we can write $S = s + U$. Now consider writing $s = s_{U} + s_{U^{perp}}$ where the subscripts denote the orthogonal projection. Then $S = s_{U^{perp}} + U$. I would like to know whether this representation, i.e., $s_{U^{perp}}$, is a uniquely defined vector.
linear-algebra projection
asked 2 days ago
user43210
153
New contributor
user43210 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Suppose $S subset mathbb R^n$ is a linear affine subspace. I know picking any $s in S$, $S- s =: U$ is a subspace and we can write $S = s + U$. Now consider writing $s = s_{U} + s_{U^{perp}}$ where the subscripts denote the orthogonal projection. Then $S = s_{U^{perp}} + U$. I would like to know whether this representation, i.e., $s_{U^{perp}}$, is a uniquely defined vector.
linear-algebra projection
asked 2 days ago
user43210
153
New contributor
user43210 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Suppose $S subset mathbb R^n$ is a linear affine subspace. I know picking any $s in S$, $S- s =: U$ is a subspace and we can write $S = s + U$. Now consider writing $s = s_{U} + s_{U^{perp}}$ where the subscripts denote the orthogonal projection. Then $S = s_{U^{perp}} + U$. I would like to know whether this representation, i.e., $s_{U^{perp}}$, is a uniquely defined vector.
linear-algebra projection
linear-algebra projection
asked 2 days ago
user43210
153
New contributor
user43210 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 2 days ago
user43210
153
New contributor
user43210 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 2 days ago
user43210
153
New contributor
user43210 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 2 days ago
user43210
153
asked 2 days ago
user43210
153
153
New contributor
user43210 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
user43210 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
user43210 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
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