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Change of basis and transformation matrix
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Maybe this question will sound silly, but I cannot formulate a complete understanding.
I have observed that the standard examples for change of basis always involves one standard basis. Usually two non-standard basis sets will be given, w.r.t the standard basis, and we are required to calculate the transformation matrix between these two. My question is
What if the standard basis is not unitary? What if we have a very different standard basis?
Will the transformation matrix depend on the choice of the coordinate system chosen for the standard basis? Like Cartesian unitary vs. polar/cylindrical?
linear-algebra vector-spaces linear-transformations tensors
asked Jul 31 at 4:20
archangel89
433321
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Maybe this question will sound silly, but I cannot formulate a complete understanding.
I have observed that the standard examples for change of basis always involves one standard basis. Usually two non-standard basis sets will be given, w.r.t the standard basis, and we are required to calculate the transformation matrix between these two. My question is
What if the standard basis is not unitary? What if we have a very different standard basis?
Will the transformation matrix depend on the choice of the coordinate system chosen for the standard basis? Like Cartesian unitary vs. polar/cylindrical?
linear-algebra vector-spaces linear-transformations tensors
asked Jul 31 at 4:20
archangel89
433321
I don't really get your question. You can calculate the transformation matrix of a map w.r.t any two basis'. It is not clear to me, what you mean by "having a very different standard basis". Different from what? I know the term "standard basis" for $mathbbR^n$ or for the polynomial ring $mathbbK[X]$. What is a "standard basis" for you? For the second question: You have to be careful. polar/cylindrical coordinates are not linear. They do not fit together with linear maps/matrix multiplication.
– Babelfish
Jul 31 at 8:00
1. Let's say we have 2 oblique basis sets. Usually these 2 sets are expressed in terms of an orthonormal standard basis [ (0,0,1),(1,0,0),(0,1,0)]. Then the Transformation matrix between these 2 oblique basis is calculated. My question is what if I change my standard basis? 2. Also can you tell me how can I determine the transformation between cartesian and polar basis vectors?
– archangel89
Jul 31 at 11:16
I think you are mixing up two concepts? You tagged your question with "linear algebra", but the transformation matrices of a change of basis in linear algebra is something else than a coordinate transformation.
– Babelfish
Jul 31 at 16:05
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Maybe this question will sound silly, but I cannot formulate a complete understanding.
I have observed that the standard examples for change of basis always involves one standard basis. Usually two non-standard basis sets will be given, w.r.t the standard basis, and we are required to calculate the transformation matrix between these two. My question is
What if the standard basis is not unitary? What if we have a very different standard basis?
Will the transformation matrix depend on the choice of the coordinate system chosen for the standard basis? Like Cartesian unitary vs. polar/cylindrical?
linear-algebra vector-spaces linear-transformations tensors
asked Jul 31 at 4:20
archangel89
433321
Maybe this question will sound silly, but I cannot formulate a complete understanding.
I have observed that the standard examples for change of basis always involves one standard basis. Usually two non-standard basis sets will be given, w.r.t the standard basis, and we are required to calculate the transformation matrix between these two. My question is
What if the standard basis is not unitary? What if we have a very different standard basis?
Will the transformation matrix depend on the choice of the coordinate system chosen for the standard basis? Like Cartesian unitary vs. polar/cylindrical?
linear-algebra vector-spaces linear-transformations tensors
asked Jul 31 at 4:20
archangel89
433321
asked Jul 31 at 4:20
archangel89
433321
asked Jul 31 at 4:20
archangel89
433321
asked Jul 31 at 4:20
archangel89
433321
433321
I don't really get your question. You can calculate the transformation matrix of a map w.r.t any two basis'. It is not clear to me, what you mean by "having a very different standard basis". Different from what? I know the term "standard basis" for $mathbbR^n$ or for the polynomial ring $mathbbK[X]$. What is a "standard basis" for you? For the second question: You have to be careful. polar/cylindrical coordinates are not linear. They do not fit together with linear maps/matrix multiplication.
– Babelfish
Jul 31 at 8:00
1. Let's say we have 2 oblique basis sets. Usually these 2 sets are expressed in terms of an orthonormal standard basis [ (0,0,1),(1,0,0),(0,1,0)]. Then the Transformation matrix between these 2 oblique basis is calculated. My question is what if I change my standard basis? 2. Also can you tell me how can I determine the transformation between cartesian and polar basis vectors?
– archangel89
Jul 31 at 11:16
I think you are mixing up two concepts? You tagged your question with "linear algebra", but the transformation matrices of a change of basis in linear algebra is something else than a coordinate transformation.
– Babelfish
Jul 31 at 16:05
add a comment |Â
I don't really get your question. You can calculate the transformation matrix of a map w.r.t any two basis'. It is not clear to me, what you mean by "having a very different standard basis". Different from what? I know the term "standard basis" for $mathbbR^n$ or for the polynomial ring $mathbbK[X]$. What is a "standard basis" for you? For the second question: You have to be careful. polar/cylindrical coordinates are not linear. They do not fit together with linear maps/matrix multiplication.
– Babelfish
Jul 31 at 8:00
1. Let's say we have 2 oblique basis sets. Usually these 2 sets are expressed in terms of an orthonormal standard basis [ (0,0,1),(1,0,0),(0,1,0)]. Then the Transformation matrix between these 2 oblique basis is calculated. My question is what if I change my standard basis? 2. Also can you tell me how can I determine the transformation between cartesian and polar basis vectors?
– archangel89
Jul 31 at 11:16
I think you are mixing up two concepts? You tagged your question with "linear algebra", but the transformation matrices of a change of basis in linear algebra is something else than a coordinate transformation.
– Babelfish
Jul 31 at 16:05
I don't really get your question. You can calculate the transformation matrix of a map w.r.t any two basis'. It is not clear to me, what you mean by "having a very different standard basis". Different from what? I know the term "standard basis" for $mathbbR^n$ or for the polynomial ring $mathbbK[X]$. What is a "standard basis" for you? For the second question: You have to be careful. polar/cylindrical coordinates are not linear. They do not fit together with linear maps/matrix multiplication.
– Babelfish
Jul 31 at 8:00
I don't really get your question. You can calculate the transformation matrix of a map w.r.t any two basis'. It is not clear to me, what you mean by "having a very different standard basis". Different from what? I know the term "standard basis" for $mathbbR^n$ or for the polynomial ring $mathbbK[X]$. What is a "standard basis" for you? For the second question: You have to be careful. polar/cylindrical coordinates are not linear. They do not fit together with linear maps/matrix multiplication.
– Babelfish
Jul 31 at 8:00
1. Let's say we have 2 oblique basis sets. Usually these 2 sets are expressed in terms of an orthonormal standard basis [ (0,0,1),(1,0,0),(0,1,0)]. Then the Transformation matrix between these 2 oblique basis is calculated. My question is what if I change my standard basis? 2. Also can you tell me how can I determine the transformation between cartesian and polar basis vectors?
– archangel89
Jul 31 at 11:16
1. Let's say we have 2 oblique basis sets. Usually these 2 sets are expressed in terms of an orthonormal standard basis [ (0,0,1),(1,0,0),(0,1,0)]. Then the Transformation matrix between these 2 oblique basis is calculated. My question is what if I change my standard basis? 2. Also can you tell me how can I determine the transformation between cartesian and polar basis vectors?
– archangel89
Jul 31 at 11:16
I think you are mixing up two concepts? You tagged your question with "linear algebra", but the transformation matrices of a change of basis in linear algebra is something else than a coordinate transformation.
– Babelfish
Jul 31 at 16:05
I think you are mixing up two concepts? You tagged your question with "linear algebra", but the transformation matrices of a change of basis in linear algebra is something else than a coordinate transformation.
– Babelfish
Jul 31 at 16:05
add a comment |Â
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I don't really get your question. You can calculate the transformation matrix of a map w.r.t any two basis'. It is not clear to me, what you mean by "having a very different standard basis". Different from what? I know the term "standard basis" for $mathbbR^n$ or for the polynomial ring $mathbbK[X]$. What is a "standard basis" for you? For the second question: You have to be careful. polar/cylindrical coordinates are not linear. They do not fit together with linear maps/matrix multiplication.
– Babelfish
Jul 31 at 8:00
1. Let's say we have 2 oblique basis sets. Usually these 2 sets are expressed in terms of an orthonormal standard basis [ (0,0,1),(1,0,0),(0,1,0)]. Then the Transformation matrix between these 2 oblique basis is calculated. My question is what if I change my standard basis? 2. Also can you tell me how can I determine the transformation between cartesian and polar basis vectors?
– archangel89
Jul 31 at 11:16
I think you are mixing up two concepts? You tagged your question with "linear algebra", but the transformation matrices of a change of basis in linear algebra is something else than a coordinate transformation.
– Babelfish
Jul 31 at 16:05