Transformation vs projection matrix, what are the differences by definition

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Is there a defined difference between the terms "projection" and "transformation" matrix? Is it that e.g., a transformation preserves the vector space of the object being transformed whereas projection can also imply projecting something into a new vector space with different basis vectors?

linear-algebra linear-transformations

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asked Jul 27 at 18:10

Sebastian

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up vote
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Is there a defined difference between the terms "projection" and "transformation" matrix? Is it that e.g., a transformation preserves the vector space of the object being transformed whereas projection can also imply projecting something into a new vector space with different basis vectors?

linear-algebra linear-transformations

share|cite|improve this question

asked Jul 27 at 18:10

Sebastian

518

add a comment | 

up vote
2
down vote

favorite

up vote
2
down vote

favorite

Is there a defined difference between the terms "projection" and "transformation" matrix? Is it that e.g., a transformation preserves the vector space of the object being transformed whereas projection can also imply projecting something into a new vector space with different basis vectors?

linear-algebra linear-transformations

share|cite|improve this question

asked Jul 27 at 18:10

Sebastian

518

Is there a defined difference between the terms "projection" and "transformation" matrix? Is it that e.g., a transformation preserves the vector space of the object being transformed whereas projection can also imply projecting something into a new vector space with different basis vectors?

linear-algebra linear-transformations

share|cite|improve this question

asked Jul 27 at 18:10

Sebastian

518

share|cite|improve this question

share|cite|improve this question

asked Jul 27 at 18:10

Sebastian

518

asked Jul 27 at 18:10

Sebastian

518

asked Jul 27 at 18:10

Sebastian

518

518

add a comment | 

add a comment | 

2 Answers
2

active

oldest

votes

up vote
1
down vote



accepted

A projection is a special kind of transformation that

• goes from one space to itself


• satisfies $$P^2=P,$$
  in the sense that applying it twice makes no difference.

Note that if the domain and codomain were unrelated, it would make no sense to apply it twice.

---

A second difference given the wording used in the question, is that a transformation is not intrinsically a matrix. A transformation can be represented by a matrix, if you have specified a basis for both the domain and codomain (which could be a different basis even if the two spaces are the same).

share|cite|improve this answer

answered Jul 27 at 18:16

Arnaud Mortier

18.6k21958

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Ah thanks, that makes sense. So, regarding the second part of your answer, an example of a transformation would be simply a scaling by a factor (scalar). Say factor a, then a*v, then v would still be in the same vector space with the same basis vectors but just scaled by a factor or a.
  – Sebastian
  Jul 27 at 18:29
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @Sebastian Yes. Regarding the comment you made on the other answer, you're wrong. A linear map is called a transformation even with different dimensions. Whereas for a projection matrix, you need a square matrix if you want to multiply it by itself (to check that $Ptimes P=P$).
  – Arnaud Mortier
  Jul 27 at 19:42
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Awesome, thanks for clearing that up!
  – Sebastian
  Jul 28 at 0:58
  

  
  
  
  

add a comment | 

up vote
2
down vote

"Transformation" can be almost anything. "Projections" map everything into a (usually) smaller subspace in which nothing changes. So a projection acts like the identity on its image. Usually this is written as $Pcirc P=P$ or $P^2=P$.

share|cite|improve this answer

answered Jul 27 at 18:18

Kusma

1,097111

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Thanks, so basically a projection matrix would be a special case of a transformation matrix. If I have a mxn dimensional matrix A and an nx1 dimensional vector v, then A is a linear transformation matrix if m=n, and probably some sort of projection otherwise
  – Sebastian
  Jul 27 at 18:26
  

  
  
  
  

add a comment | 

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2 Answers
2

active

oldest

votes

2 Answers
2

active

oldest

votes

active

oldest

votes

active

oldest

votes

up vote
1
down vote



accepted

A projection is a special kind of transformation that

• goes from one space to itself


• satisfies $$P^2=P,$$
  in the sense that applying it twice makes no difference.

Note that if the domain and codomain were unrelated, it would make no sense to apply it twice.

---

A second difference given the wording used in the question, is that a transformation is not intrinsically a matrix. A transformation can be represented by a matrix, if you have specified a basis for both the domain and codomain (which could be a different basis even if the two spaces are the same).

share|cite|improve this answer

answered Jul 27 at 18:16

Arnaud Mortier

18.6k21958

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Ah thanks, that makes sense. So, regarding the second part of your answer, an example of a transformation would be simply a scaling by a factor (scalar). Say factor a, then a*v, then v would still be in the same vector space with the same basis vectors but just scaled by a factor or a.
  – Sebastian
  Jul 27 at 18:29
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @Sebastian Yes. Regarding the comment you made on the other answer, you're wrong. A linear map is called a transformation even with different dimensions. Whereas for a projection matrix, you need a square matrix if you want to multiply it by itself (to check that $Ptimes P=P$).
  – Arnaud Mortier
  Jul 27 at 19:42
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Awesome, thanks for clearing that up!
  – Sebastian
  Jul 28 at 0:58
  

  
  
  
  

add a comment | 

up vote
1
down vote



accepted

A projection is a special kind of transformation that

• goes from one space to itself


• satisfies $$P^2=P,$$
  in the sense that applying it twice makes no difference.

Note that if the domain and codomain were unrelated, it would make no sense to apply it twice.

---

A second difference given the wording used in the question, is that a transformation is not intrinsically a matrix. A transformation can be represented by a matrix, if you have specified a basis for both the domain and codomain (which could be a different basis even if the two spaces are the same).

share|cite|improve this answer

answered Jul 27 at 18:16

Arnaud Mortier

18.6k21958

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Ah thanks, that makes sense. So, regarding the second part of your answer, an example of a transformation would be simply a scaling by a factor (scalar). Say factor a, then a*v, then v would still be in the same vector space with the same basis vectors but just scaled by a factor or a.
  – Sebastian
  Jul 27 at 18:29
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @Sebastian Yes. Regarding the comment you made on the other answer, you're wrong. A linear map is called a transformation even with different dimensions. Whereas for a projection matrix, you need a square matrix if you want to multiply it by itself (to check that $Ptimes P=P$).
  – Arnaud Mortier
  Jul 27 at 19:42
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Awesome, thanks for clearing that up!
  – Sebastian
  Jul 28 at 0:58
  

  
  
  
  

add a comment | 

up vote
1
down vote



accepted

up vote
1
down vote



accepted

A projection is a special kind of transformation that

• goes from one space to itself


• satisfies $$P^2=P,$$
  in the sense that applying it twice makes no difference.

Note that if the domain and codomain were unrelated, it would make no sense to apply it twice.

---

A second difference given the wording used in the question, is that a transformation is not intrinsically a matrix. A transformation can be represented by a matrix, if you have specified a basis for both the domain and codomain (which could be a different basis even if the two spaces are the same).

share|cite|improve this answer

answered Jul 27 at 18:16

Arnaud Mortier

18.6k21958

A projection is a special kind of transformation that

• goes from one space to itself


• satisfies $$P^2=P,$$
  in the sense that applying it twice makes no difference.

Note that if the domain and codomain were unrelated, it would make no sense to apply it twice.

---

A second difference given the wording used in the question, is that a transformation is not intrinsically a matrix. A transformation can be represented by a matrix, if you have specified a basis for both the domain and codomain (which could be a different basis even if the two spaces are the same).

share|cite|improve this answer

answered Jul 27 at 18:16

Arnaud Mortier

18.6k21958

share|cite|improve this answer

share|cite|improve this answer

answered Jul 27 at 18:16

Arnaud Mortier

18.6k21958

answered Jul 27 at 18:16

Arnaud Mortier

18.6k21958

answered Jul 27 at 18:16

Arnaud Mortier

18.6k21958

18.6k21958

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Ah thanks, that makes sense. So, regarding the second part of your answer, an example of a transformation would be simply a scaling by a factor (scalar). Say factor a, then a*v, then v would still be in the same vector space with the same basis vectors but just scaled by a factor or a.
  – Sebastian
  Jul 27 at 18:29
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @Sebastian Yes. Regarding the comment you made on the other answer, you're wrong. A linear map is called a transformation even with different dimensions. Whereas for a projection matrix, you need a square matrix if you want to multiply it by itself (to check that $Ptimes P=P$).
  – Arnaud Mortier
  Jul 27 at 19:42
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Awesome, thanks for clearing that up!
  – Sebastian
  Jul 28 at 0:58
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Ah thanks, that makes sense. So, regarding the second part of your answer, an example of a transformation would be simply a scaling by a factor (scalar). Say factor a, then a*v, then v would still be in the same vector space with the same basis vectors but just scaled by a factor or a.
  – Sebastian
  Jul 27 at 18:29
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @Sebastian Yes. Regarding the comment you made on the other answer, you're wrong. A linear map is called a transformation even with different dimensions. Whereas for a projection matrix, you need a square matrix if you want to multiply it by itself (to check that $Ptimes P=P$).
  – Arnaud Mortier
  Jul 27 at 19:42
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Awesome, thanks for clearing that up!
  – Sebastian
  Jul 28 at 0:58
  

  
  
  
  

Ah thanks, that makes sense. So, regarding the second part of your answer, an example of a transformation would be simply a scaling by a factor (scalar). Say factor a, then a*v, then v would still be in the same vector space with the same basis vectors but just scaled by a factor or a.
– Sebastian
Jul 27 at 18:29

Ah thanks, that makes sense. So, regarding the second part of your answer, an example of a transformation would be simply a scaling by a factor (scalar). Say factor a, then a*v, then v would still be in the same vector space with the same basis vectors but just scaled by a factor or a.
– Sebastian
Jul 27 at 18:29

@Sebastian Yes. Regarding the comment you made on the other answer, you're wrong. A linear map is called a transformation even with different dimensions. Whereas for a projection matrix, you need a square matrix if you want to multiply it by itself (to check that $Ptimes P=P$).
– Arnaud Mortier
Jul 27 at 19:42

@Sebastian Yes. Regarding the comment you made on the other answer, you're wrong. A linear map is called a transformation even with different dimensions. Whereas for a projection matrix, you need a square matrix if you want to multiply it by itself (to check that $Ptimes P=P$).
– Arnaud Mortier
Jul 27 at 19:42

Awesome, thanks for clearing that up!
– Sebastian
Jul 28 at 0:58

Awesome, thanks for clearing that up!
– Sebastian
Jul 28 at 0:58

add a comment | 

up vote
2
down vote

"Transformation" can be almost anything. "Projections" map everything into a (usually) smaller subspace in which nothing changes. So a projection acts like the identity on its image. Usually this is written as $Pcirc P=P$ or $P^2=P$.

share|cite|improve this answer

answered Jul 27 at 18:18

Kusma

1,097111

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Thanks, so basically a projection matrix would be a special case of a transformation matrix. If I have a mxn dimensional matrix A and an nx1 dimensional vector v, then A is a linear transformation matrix if m=n, and probably some sort of projection otherwise
  – Sebastian
  Jul 27 at 18:26
  

  
  
  
  

add a comment | 

up vote
2
down vote

"Transformation" can be almost anything. "Projections" map everything into a (usually) smaller subspace in which nothing changes. So a projection acts like the identity on its image. Usually this is written as $Pcirc P=P$ or $P^2=P$.

share|cite|improve this answer

answered Jul 27 at 18:18

Kusma

1,097111

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Thanks, so basically a projection matrix would be a special case of a transformation matrix. If I have a mxn dimensional matrix A and an nx1 dimensional vector v, then A is a linear transformation matrix if m=n, and probably some sort of projection otherwise
  – Sebastian
  Jul 27 at 18:26
  

  
  
  
  

add a comment | 

up vote
2
down vote

up vote
2
down vote

"Transformation" can be almost anything. "Projections" map everything into a (usually) smaller subspace in which nothing changes. So a projection acts like the identity on its image. Usually this is written as $Pcirc P=P$ or $P^2=P$.

share|cite|improve this answer

answered Jul 27 at 18:18

Kusma

1,097111

"Transformation" can be almost anything. "Projections" map everything into a (usually) smaller subspace in which nothing changes. So a projection acts like the identity on its image. Usually this is written as $Pcirc P=P$ or $P^2=P$.

share|cite|improve this answer

answered Jul 27 at 18:18

Kusma

1,097111

share|cite|improve this answer

share|cite|improve this answer

answered Jul 27 at 18:18

Kusma

1,097111

answered Jul 27 at 18:18

Kusma

1,097111

answered Jul 27 at 18:18

Kusma

1,097111

1,097111

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Thanks, so basically a projection matrix would be a special case of a transformation matrix. If I have a mxn dimensional matrix A and an nx1 dimensional vector v, then A is a linear transformation matrix if m=n, and probably some sort of projection otherwise
  – Sebastian
  Jul 27 at 18:26
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Thanks, so basically a projection matrix would be a special case of a transformation matrix. If I have a mxn dimensional matrix A and an nx1 dimensional vector v, then A is a linear transformation matrix if m=n, and probably some sort of projection otherwise
  – Sebastian
  Jul 27 at 18:26
  

  
  
  
  

Thanks, so basically a projection matrix would be a special case of a transformation matrix. If I have a mxn dimensional matrix A and an nx1 dimensional vector v, then A is a linear transformation matrix if m=n, and probably some sort of projection otherwise
– Sebastian
Jul 27 at 18:26

Thanks, so basically a projection matrix would be a special case of a transformation matrix. If I have a mxn dimensional matrix A and an nx1 dimensional vector v, then A is a linear transformation matrix if m=n, and probably some sort of projection otherwise
– Sebastian
Jul 27 at 18:26

add a comment | 

 

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