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Definition of span
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On an old midterm exam, my professor requested the students prove that
The span of $S$ (where $S$ is a subset of a vector space $V$) is equal to all vectors that can be expressed as linear combinations of the elements in $S$.
Does this make any sense? He's requesting we show that the span of $S$ equals what I believe to be the definition of span. Is there possibly some other definition of span that I should be aware of?
linear-algebra vector-spaces definition
edited 1 hour ago
José Carlos Santos
128k17103191
asked 1 hour ago
DavidS
161110
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up vote
2
down vote
favorite
On an old midterm exam, my professor requested the students prove that
The span of $S$ (where $S$ is a subset of a vector space $V$) is equal to all vectors that can be expressed as linear combinations of the elements in $S$.
Does this make any sense? He's requesting we show that the span of $S$ equals what I believe to be the definition of span. Is there possibly some other definition of span that I should be aware of?
linear-algebra vector-spaces definition
edited 1 hour ago
José Carlos Santos
128k17103191
asked 1 hour ago
DavidS
161110
And of course the true insight is that one proves the equivalence of all these properties and thereby is motivated to define the notion of span (as any and all of these) in the first place ...
– Hagen von Eitzen
1 hour ago
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
On an old midterm exam, my professor requested the students prove that
The span of $S$ (where $S$ is a subset of a vector space $V$) is equal to all vectors that can be expressed as linear combinations of the elements in $S$.
Does this make any sense? He's requesting we show that the span of $S$ equals what I believe to be the definition of span. Is there possibly some other definition of span that I should be aware of?
linear-algebra vector-spaces definition
edited 1 hour ago
José Carlos Santos
128k17103191
asked 1 hour ago
DavidS
161110
On an old midterm exam, my professor requested the students prove that
The span of $S$ (where $S$ is a subset of a vector space $V$) is equal to all vectors that can be expressed as linear combinations of the elements in $S$.
Does this make any sense? He's requesting we show that the span of $S$ equals what I believe to be the definition of span. Is there possibly some other definition of span that I should be aware of?
linear-algebra vector-spaces definition
linear-algebra vector-spaces definition
edited 1 hour ago
José Carlos Santos
128k17103191
asked 1 hour ago
DavidS
161110
edited 1 hour ago
José Carlos Santos
128k17103191
asked 1 hour ago
DavidS
161110
edited 1 hour ago
José Carlos Santos
128k17103191
edited 1 hour ago
José Carlos Santos
128k17103191
edited 1 hour ago
José Carlos Santos
128k17103191
128k17103191
asked 1 hour ago
DavidS
161110
asked 1 hour ago
DavidS
161110
asked 1 hour ago
DavidS
161110
161110
And of course the true insight is that one proves the equivalence of all these properties and thereby is motivated to define the notion of span (as any and all of these) in the first place ...
– Hagen von Eitzen
1 hour ago
add a comment |Â
And of course the true insight is that one proves the equivalence of all these properties and thereby is motivated to define the notion of span (as any and all of these) in the first place ...
– Hagen von Eitzen
1 hour ago
And of course the true insight is that one proves the equivalence of all these properties and thereby is motivated to define the notion of span (as any and all of these) in the first place ...
– Hagen von Eitzen
1 hour ago
And of course the true insight is that one proves the equivalence of all these properties and thereby is motivated to define the notion of span (as any and all of these) in the first place ...
– Hagen von Eitzen
1 hour ago
add a comment |Â
3 Answers
3
active
oldest
votes
up vote
3
down vote
Yes, there is another definition: let$$mathcalW=bigcapleftWsubset V,middle.$$Now, define $operatornamespanS=bigcap_Winmathcal WW$.
answered 1 hour ago
José Carlos Santos
128k17103191
add a comment |Â
up vote
3
down vote
Perhaps the definition of span that your professor is using is: The smallest vector space generated by the vectors in the spanning set.
answered 1 hour ago
ervx
9,81331337
add a comment |Â
up vote
3
down vote
You can define $textspan (S)$ to be the smallest vector subspace containing $S$, or equivalently the intersection all vector subspaces containing $S$. Such a definition is very common in algebra.
answered 1 hour ago
Foobaz John
18.9k41245
add a comment |Â
3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
Yes, there is another definition: let$$mathcalW=bigcapleftWsubset V,middle.$$Now, define $operatornamespanS=bigcap_Winmathcal WW$.
answered 1 hour ago
José Carlos Santos
128k17103191
add a comment |Â
up vote
3
down vote
Yes, there is another definition: let$$mathcalW=bigcapleftWsubset V,middle.$$Now, define $operatornamespanS=bigcap_Winmathcal WW$.
answered 1 hour ago
José Carlos Santos
128k17103191
add a comment |Â
up vote
3
down vote
up vote
3
down vote
Yes, there is another definition: let$$mathcalW=bigcapleftWsubset V,middle.$$Now, define $operatornamespanS=bigcap_Winmathcal WW$.
answered 1 hour ago
José Carlos Santos
128k17103191
Yes, there is another definition: let$$mathcalW=bigcapleftWsubset V,middle.$$Now, define $operatornamespanS=bigcap_Winmathcal WW$.
answered 1 hour ago
José Carlos Santos
128k17103191
answered 1 hour ago
José Carlos Santos
128k17103191
answered 1 hour ago
José Carlos Santos
128k17103191
answered 1 hour ago
José Carlos Santos
128k17103191
128k17103191
add a comment |Â
add a comment |Â
up vote
3
down vote
Perhaps the definition of span that your professor is using is: The smallest vector space generated by the vectors in the spanning set.
answered 1 hour ago
ervx
9,81331337
add a comment |Â
up vote
3
down vote
Perhaps the definition of span that your professor is using is: The smallest vector space generated by the vectors in the spanning set.
answered 1 hour ago
ervx
9,81331337
add a comment |Â
up vote
3
down vote
up vote
3
down vote
Perhaps the definition of span that your professor is using is: The smallest vector space generated by the vectors in the spanning set.
answered 1 hour ago
ervx
9,81331337
Perhaps the definition of span that your professor is using is: The smallest vector space generated by the vectors in the spanning set.
answered 1 hour ago
ervx
9,81331337
answered 1 hour ago
ervx
9,81331337
answered 1 hour ago
ervx
9,81331337
answered 1 hour ago
ervx
9,81331337
9,81331337
add a comment |Â
add a comment |Â
up vote
3
down vote
You can define $textspan (S)$ to be the smallest vector subspace containing $S$, or equivalently the intersection all vector subspaces containing $S$. Such a definition is very common in algebra.
answered 1 hour ago
Foobaz John
18.9k41245
add a comment |Â
up vote
3
down vote
You can define $textspan (S)$ to be the smallest vector subspace containing $S$, or equivalently the intersection all vector subspaces containing $S$. Such a definition is very common in algebra.
answered 1 hour ago
Foobaz John
18.9k41245
add a comment |Â
up vote
3
down vote
up vote
3
down vote
You can define $textspan (S)$ to be the smallest vector subspace containing $S$, or equivalently the intersection all vector subspaces containing $S$. Such a definition is very common in algebra.
answered 1 hour ago
Foobaz John
18.9k41245
You can define $textspan (S)$ to be the smallest vector subspace containing $S$, or equivalently the intersection all vector subspaces containing $S$. Such a definition is very common in algebra.
answered 1 hour ago
Foobaz John
18.9k41245
answered 1 hour ago
Foobaz John
18.9k41245
answered 1 hour ago
Foobaz John
18.9k41245
answered 1 hour ago
Foobaz John
18.9k41245
18.9k41245
add a comment |Â
add a comment |Â
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And of course the true insight is that one proves the equivalence of all these properties and thereby is motivated to define the notion of span (as any and all of these) in the first place ...
– Hagen von Eitzen
1 hour ago