Word for the dimension of the vector space in which a vector lives?
The following issue comes up whenever I teach linear algebra: I want to have a quick way to say that a vector $(x,y,z)$ is in $mathbb{R}^3$. I am tempted to say that it has "length $3$". But then some student interprets this as saying that $sqrt{x^2+y^2+z^2} = 3$. Is there some word other than "length" I could introduce and use to consistently refer to the number of coordinates in a vector? (To point out that I am not nuts here, the commands to get this number in Mathematica and MATLAB are Length and length().)
linear-algebra vocabulary
asked Dec 9 at 15:42
David E Speyer
2,295920
|
show 2 more comments
The following issue comes up whenever I teach linear algebra: I want to have a quick way to say that a vector $(x,y,z)$ is in $mathbb{R}^3$. I am tempted to say that it has "length $3$". But then some student interprets this as saying that $sqrt{x^2+y^2+z^2} = 3$. Is there some word other than "length" I could introduce and use to consistently refer to the number of coordinates in a vector? (To point out that I am not nuts here, the commands to get this number in Mathematica and MATLAB are Length and length().)
linear-algebra vocabulary
asked Dec 9 at 15:42
David E Speyer
2,295920
3
Would "dimension" do?
– Jasper
Dec 9 at 16:22
2
This notion is extrinsic; there is no way to look at an object and tell 'the' vector space it belongs to (and one has stuff like $mathbb{R}^{2n} simeq mathbb{C}^n$). But seconded, maybe 'of real dimension 3' or 'real 3-tuple' might do.
– Vandermonde
Dec 9 at 16:29
I would also use dimension, but if the vector is known to sit inside some proper subspace I might also use "ambient dimension" to indicate how many components it has.
– Adam
Dec 9 at 16:44
1
@user683: $n$-vector has the unfortunate collision with certain physics terminology.
– Willie Wong
Dec 10 at 15:02
1
More to the point of the question itself: which kind of linear algebra course are you teaching? Is this an abstract linear algebra course, or an engineering one? What is your main perspective in terms of identifying an $n$-dimensional vector space over the field $mathbb{F}$ with the space of $n$-tuples $mathbb{F}^n$? I ask because the programming conventions are implicitly fixing a basis and always identifying vectors with tuples, hence thelength().
– Willie Wong
Dec 10 at 15:05
|
show 2 more comments
The following issue comes up whenever I teach linear algebra: I want to have a quick way to say that a vector $(x,y,z)$ is in $mathbb{R}^3$. I am tempted to say that it has "length $3$". But then some student interprets this as saying that $sqrt{x^2+y^2+z^2} = 3$. Is there some word other than "length" I could introduce and use to consistently refer to the number of coordinates in a vector? (To point out that I am not nuts here, the commands to get this number in Mathematica and MATLAB are Length and length().)
linear-algebra vocabulary
asked Dec 9 at 15:42
David E Speyer
2,295920
The following issue comes up whenever I teach linear algebra: I want to have a quick way to say that a vector $(x,y,z)$ is in $mathbb{R}^3$. I am tempted to say that it has "length $3$". But then some student interprets this as saying that $sqrt{x^2+y^2+z^2} = 3$. Is there some word other than "length" I could introduce and use to consistently refer to the number of coordinates in a vector? (To point out that I am not nuts here, the commands to get this number in Mathematica and MATLAB are Length and length().)
linear-algebra vocabulary
linear-algebra vocabulary
asked Dec 9 at 15:42
David E Speyer
2,295920
asked Dec 9 at 15:42
David E Speyer
2,295920
edited Dec 9 at 16:38
asked Dec 9 at 15:42
David E Speyer
2,295920
asked Dec 9 at 15:42
David E Speyer
2,295920
asked Dec 9 at 15:42
David E Speyer
2,295920
2,295920
3
Would "dimension" do?
– Jasper
Dec 9 at 16:22
2
This notion is extrinsic; there is no way to look at an object and tell 'the' vector space it belongs to (and one has stuff like $mathbb{R}^{2n} simeq mathbb{C}^n$). But seconded, maybe 'of real dimension 3' or 'real 3-tuple' might do.
– Vandermonde
Dec 9 at 16:29
I would also use dimension, but if the vector is known to sit inside some proper subspace I might also use "ambient dimension" to indicate how many components it has.
– Adam
Dec 9 at 16:44
1
@user683: $n$-vector has the unfortunate collision with certain physics terminology.
– Willie Wong
Dec 10 at 15:02
1
More to the point of the question itself: which kind of linear algebra course are you teaching? Is this an abstract linear algebra course, or an engineering one? What is your main perspective in terms of identifying an $n$-dimensional vector space over the field $mathbb{F}$ with the space of $n$-tuples $mathbb{F}^n$? I ask because the programming conventions are implicitly fixing a basis and always identifying vectors with tuples, hence thelength().
– Willie Wong
Dec 10 at 15:05
|
show 2 more comments
3
Would "dimension" do?
– Jasper
Dec 9 at 16:22
2
This notion is extrinsic; there is no way to look at an object and tell 'the' vector space it belongs to (and one has stuff like $mathbb{R}^{2n} simeq mathbb{C}^n$). But seconded, maybe 'of real dimension 3' or 'real 3-tuple' might do.
– Vandermonde
Dec 9 at 16:29
I would also use dimension, but if the vector is known to sit inside some proper subspace I might also use "ambient dimension" to indicate how many components it has.
– Adam
Dec 9 at 16:44
1
@user683: $n$-vector has the unfortunate collision with certain physics terminology.
– Willie Wong
Dec 10 at 15:02
1
More to the point of the question itself: which kind of linear algebra course are you teaching? Is this an abstract linear algebra course, or an engineering one? What is your main perspective in terms of identifying an $n$-dimensional vector space over the field $mathbb{F}$ with the space of $n$-tuples $mathbb{F}^n$? I ask because the programming conventions are implicitly fixing a basis and always identifying vectors with tuples, hence thelength().
– Willie Wong
Dec 10 at 15:05
3
3
Would "dimension" do?
– Jasper
Dec 9 at 16:22
Would "dimension" do?
– Jasper
Dec 9 at 16:22
2
2
This notion is extrinsic; there is no way to look at an object and tell 'the' vector space it belongs to (and one has stuff like $mathbb{R}^{2n} simeq mathbb{C}^n$). But seconded, maybe 'of real dimension 3' or 'real 3-tuple' might do.
– Vandermonde
Dec 9 at 16:29
This notion is extrinsic; there is no way to look at an object and tell 'the' vector space it belongs to (and one has stuff like $mathbb{R}^{2n} simeq mathbb{C}^n$). But seconded, maybe 'of real dimension 3' or 'real 3-tuple' might do.
– Vandermonde
Dec 9 at 16:29
I would also use dimension, but if the vector is known to sit inside some proper subspace I might also use "ambient dimension" to indicate how many components it has.
– Adam
Dec 9 at 16:44
I would also use dimension, but if the vector is known to sit inside some proper subspace I might also use "ambient dimension" to indicate how many components it has.
– Adam
Dec 9 at 16:44
1
1
@user683: $n$-vector has the unfortunate collision with certain physics terminology.
– Willie Wong
Dec 10 at 15:02
@user683: $n$-vector has the unfortunate collision with certain physics terminology.
– Willie Wong
Dec 10 at 15:02
1
1
More to the point of the question itself: which kind of linear algebra course are you teaching? Is this an abstract linear algebra course, or an engineering one? What is your main perspective in terms of identifying an $n$-dimensional vector space over the field $mathbb{F}$ with the space of $n$-tuples $mathbb{F}^n$? I ask because the programming conventions are implicitly fixing a basis and always identifying vectors with tuples, hence the
length().– Willie Wong
Dec 10 at 15:05
More to the point of the question itself: which kind of linear algebra course are you teaching? Is this an abstract linear algebra course, or an engineering one? What is your main perspective in terms of identifying an $n$-dimensional vector space over the field $mathbb{F}$ with the space of $n$-tuples $mathbb{F}^n$? I ask because the programming conventions are implicitly fixing a basis and always identifying vectors with tuples, hence the
length().– Willie Wong
Dec 10 at 15:05
|
show 2 more comments
3 Answers
3
active
oldest
votes
If all your spaces are $mathbb{R}^n$ then you can say n-dimensional (three-dimensional for $mathbb{R}^3$). However the space of matrices $displaystyle left(begin{array}{cc}a&b\c&dend{array}right)$ ($a,b,c,dinmathbb{C}$) is four-dimensional over $mathbb{C}$, eight-dimensional over $mathbb{R}$ and infinite-dimensional over $mathbb{Q}$. So be clear what the base field is. Alternatively you could just write $(a,b,c)inleft(mathbb{R}^3,+,cdotright)$ or more briefly $(a,b,c)inmathbb{R}^3$.
answered Dec 9 at 17:45
BPP
579316
add a comment |
Along with "dimension", you could also use "component".
A vector in three dimensions has three components.
answered Dec 9 at 21:59
robphy
3713
add a comment |
I say "This is a 3D vector" or "This is a 7D vector".
answered Dec 9 at 16:34
Steven Gubkin
8,14312248
I feel compelled to comment that this is used in a social context in which we are working on some problem involving some vectors in $mathbb{R}^n$, where these vectors are being thought of as lists of numbers. I certainly wouldn't use this terminology if I was talking about a vector living in an arbitrary vector space.
– Steven Gubkin
Dec 11 at 1:41
add a comment |
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3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
If all your spaces are $mathbb{R}^n$ then you can say n-dimensional (three-dimensional for $mathbb{R}^3$). However the space of matrices $displaystyle left(begin{array}{cc}a&b\c&dend{array}right)$ ($a,b,c,dinmathbb{C}$) is four-dimensional over $mathbb{C}$, eight-dimensional over $mathbb{R}$ and infinite-dimensional over $mathbb{Q}$. So be clear what the base field is. Alternatively you could just write $(a,b,c)inleft(mathbb{R}^3,+,cdotright)$ or more briefly $(a,b,c)inmathbb{R}^3$.
answered Dec 9 at 17:45
BPP
579316
add a comment |
If all your spaces are $mathbb{R}^n$ then you can say n-dimensional (three-dimensional for $mathbb{R}^3$). However the space of matrices $displaystyle left(begin{array}{cc}a&b\c&dend{array}right)$ ($a,b,c,dinmathbb{C}$) is four-dimensional over $mathbb{C}$, eight-dimensional over $mathbb{R}$ and infinite-dimensional over $mathbb{Q}$. So be clear what the base field is. Alternatively you could just write $(a,b,c)inleft(mathbb{R}^3,+,cdotright)$ or more briefly $(a,b,c)inmathbb{R}^3$.
answered Dec 9 at 17:45
BPP
579316
add a comment |
If all your spaces are $mathbb{R}^n$ then you can say n-dimensional (three-dimensional for $mathbb{R}^3$). However the space of matrices $displaystyle left(begin{array}{cc}a&b\c&dend{array}right)$ ($a,b,c,dinmathbb{C}$) is four-dimensional over $mathbb{C}$, eight-dimensional over $mathbb{R}$ and infinite-dimensional over $mathbb{Q}$. So be clear what the base field is. Alternatively you could just write $(a,b,c)inleft(mathbb{R}^3,+,cdotright)$ or more briefly $(a,b,c)inmathbb{R}^3$.
answered Dec 9 at 17:45
BPP
579316
If all your spaces are $mathbb{R}^n$ then you can say n-dimensional (three-dimensional for $mathbb{R}^3$). However the space of matrices $displaystyle left(begin{array}{cc}a&b\c&dend{array}right)$ ($a,b,c,dinmathbb{C}$) is four-dimensional over $mathbb{C}$, eight-dimensional over $mathbb{R}$ and infinite-dimensional over $mathbb{Q}$. So be clear what the base field is. Alternatively you could just write $(a,b,c)inleft(mathbb{R}^3,+,cdotright)$ or more briefly $(a,b,c)inmathbb{R}^3$.
answered Dec 9 at 17:45
BPP
579316
answered Dec 9 at 17:45
BPP
579316
answered Dec 9 at 17:45
BPP
579316
answered Dec 9 at 17:45
BPP
579316
579316
add a comment |
add a comment |
Along with "dimension", you could also use "component".
A vector in three dimensions has three components.
answered Dec 9 at 21:59
robphy
3713
add a comment |
Along with "dimension", you could also use "component".
A vector in three dimensions has three components.
answered Dec 9 at 21:59
robphy
3713
add a comment |
Along with "dimension", you could also use "component".
A vector in three dimensions has three components.
answered Dec 9 at 21:59
robphy
3713
Along with "dimension", you could also use "component".
A vector in three dimensions has three components.
answered Dec 9 at 21:59
robphy
3713
answered Dec 9 at 21:59
robphy
3713
answered Dec 9 at 21:59
robphy
3713
answered Dec 9 at 21:59
robphy
3713
3713
add a comment |
add a comment |
I say "This is a 3D vector" or "This is a 7D vector".
answered Dec 9 at 16:34
Steven Gubkin
8,14312248
I feel compelled to comment that this is used in a social context in which we are working on some problem involving some vectors in $mathbb{R}^n$, where these vectors are being thought of as lists of numbers. I certainly wouldn't use this terminology if I was talking about a vector living in an arbitrary vector space.
– Steven Gubkin
Dec 11 at 1:41
add a comment |
I say "This is a 3D vector" or "This is a 7D vector".
answered Dec 9 at 16:34
Steven Gubkin
8,14312248
I feel compelled to comment that this is used in a social context in which we are working on some problem involving some vectors in $mathbb{R}^n$, where these vectors are being thought of as lists of numbers. I certainly wouldn't use this terminology if I was talking about a vector living in an arbitrary vector space.
– Steven Gubkin
Dec 11 at 1:41
add a comment |
I say "This is a 3D vector" or "This is a 7D vector".
answered Dec 9 at 16:34
Steven Gubkin
8,14312248
I say "This is a 3D vector" or "This is a 7D vector".
answered Dec 9 at 16:34
Steven Gubkin
8,14312248
answered Dec 9 at 16:34
Steven Gubkin
8,14312248
answered Dec 9 at 16:34
Steven Gubkin
8,14312248
answered Dec 9 at 16:34
Steven Gubkin
8,14312248
8,14312248
I feel compelled to comment that this is used in a social context in which we are working on some problem involving some vectors in $mathbb{R}^n$, where these vectors are being thought of as lists of numbers. I certainly wouldn't use this terminology if I was talking about a vector living in an arbitrary vector space.
– Steven Gubkin
Dec 11 at 1:41
add a comment |
I feel compelled to comment that this is used in a social context in which we are working on some problem involving some vectors in $mathbb{R}^n$, where these vectors are being thought of as lists of numbers. I certainly wouldn't use this terminology if I was talking about a vector living in an arbitrary vector space.
– Steven Gubkin
Dec 11 at 1:41
I feel compelled to comment that this is used in a social context in which we are working on some problem involving some vectors in $mathbb{R}^n$, where these vectors are being thought of as lists of numbers. I certainly wouldn't use this terminology if I was talking about a vector living in an arbitrary vector space.
– Steven Gubkin
Dec 11 at 1:41
I feel compelled to comment that this is used in a social context in which we are working on some problem involving some vectors in $mathbb{R}^n$, where these vectors are being thought of as lists of numbers. I certainly wouldn't use this terminology if I was talking about a vector living in an arbitrary vector space.
– Steven Gubkin
Dec 11 at 1:41
add a comment |
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3
Would "dimension" do?
– Jasper
Dec 9 at 16:22
2
This notion is extrinsic; there is no way to look at an object and tell 'the' vector space it belongs to (and one has stuff like $mathbb{R}^{2n} simeq mathbb{C}^n$). But seconded, maybe 'of real dimension 3' or 'real 3-tuple' might do.
– Vandermonde
Dec 9 at 16:29
I would also use dimension, but if the vector is known to sit inside some proper subspace I might also use "ambient dimension" to indicate how many components it has.
– Adam
Dec 9 at 16:44
1
@user683: $n$-vector has the unfortunate collision with certain physics terminology.
– Willie Wong
Dec 10 at 15:02
1
More to the point of the question itself: which kind of linear algebra course are you teaching? Is this an abstract linear algebra course, or an engineering one? What is your main perspective in terms of identifying an $n$-dimensional vector space over the field $mathbb{F}$ with the space of $n$-tuples $mathbb{F}^n$? I ask because the programming conventions are implicitly fixing a basis and always identifying vectors with tuples, hence the
length().– Willie Wong
Dec 10 at 15:05