The Universal Mapping Property of a free vector space.
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Definition: Let $X$ be a non-empty set. A free vector space on $X$ is a pair $(V,i)$ consisting of a vector space $V$ and a function $i:Xto V$ satisfying the following universal mapping property.
I have to prove statement; Let $(V,i)$ be a free vector space on a non-empty set $X$.
Given a vector space $U$ and a function $j:Xto U$, show that $(U,j)$ is a free vector space on $X$ if and only if there is a unique linear map $f:Uto V$ such that $fcirc j=i$.
Proof $(to)$ Clearly, by the UMP.
$(leftarrow)$ Assume that there is a unique linear map $f:Uto V$ such that $fcirc g=i$, we have to show that $(U,j)$ is a free vector space on $X$.
Let $W$ be a vector space and a function $q:Xto W$, since $(V,i)$ be a free vector space on a non-empty set $X$ , by the UMP there is a unique linear map $d:Vto W$.
I just need to show that $T=fcirc d$ is unique .
I have some question , $f$ and $d$ are unique then $T$ is unique ? (i think not true) or how to show by uniqueness.
linear-algebra multilinear-algebra universal-property
asked Nov 21 at 15:42
kim wenasa
1
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0
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Definition: Let $X$ be a non-empty set. A free vector space on $X$ is a pair $(V,i)$ consisting of a vector space $V$ and a function $i:Xto V$ satisfying the following universal mapping property.
I have to prove statement; Let $(V,i)$ be a free vector space on a non-empty set $X$.
Given a vector space $U$ and a function $j:Xto U$, show that $(U,j)$ is a free vector space on $X$ if and only if there is a unique linear map $f:Uto V$ such that $fcirc j=i$.
Proof $(to)$ Clearly, by the UMP.
$(leftarrow)$ Assume that there is a unique linear map $f:Uto V$ such that $fcirc g=i$, we have to show that $(U,j)$ is a free vector space on $X$.
Let $W$ be a vector space and a function $q:Xto W$, since $(V,i)$ be a free vector space on a non-empty set $X$ , by the UMP there is a unique linear map $d:Vto W$.
I just need to show that $T=fcirc d$ is unique .
I have some question , $f$ and $d$ are unique then $T$ is unique ? (i think not true) or how to show by uniqueness.
linear-algebra multilinear-algebra universal-property
asked Nov 21 at 15:42
kim wenasa
1
You forgot to state the universal property in question. Of course it's easy to guess what it must be, but...
– David C. Ullrich
Nov 22 at 17:34
add a comment |
up vote
0
down vote
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up vote
0
down vote
favorite
Definition: Let $X$ be a non-empty set. A free vector space on $X$ is a pair $(V,i)$ consisting of a vector space $V$ and a function $i:Xto V$ satisfying the following universal mapping property.
I have to prove statement; Let $(V,i)$ be a free vector space on a non-empty set $X$.
Given a vector space $U$ and a function $j:Xto U$, show that $(U,j)$ is a free vector space on $X$ if and only if there is a unique linear map $f:Uto V$ such that $fcirc j=i$.
Proof $(to)$ Clearly, by the UMP.
$(leftarrow)$ Assume that there is a unique linear map $f:Uto V$ such that $fcirc g=i$, we have to show that $(U,j)$ is a free vector space on $X$.
Let $W$ be a vector space and a function $q:Xto W$, since $(V,i)$ be a free vector space on a non-empty set $X$ , by the UMP there is a unique linear map $d:Vto W$.
I just need to show that $T=fcirc d$ is unique .
I have some question , $f$ and $d$ are unique then $T$ is unique ? (i think not true) or how to show by uniqueness.
linear-algebra multilinear-algebra universal-property
asked Nov 21 at 15:42
kim wenasa
1
Definition: Let $X$ be a non-empty set. A free vector space on $X$ is a pair $(V,i)$ consisting of a vector space $V$ and a function $i:Xto V$ satisfying the following universal mapping property.
I have to prove statement; Let $(V,i)$ be a free vector space on a non-empty set $X$.
Given a vector space $U$ and a function $j:Xto U$, show that $(U,j)$ is a free vector space on $X$ if and only if there is a unique linear map $f:Uto V$ such that $fcirc j=i$.
Proof $(to)$ Clearly, by the UMP.
$(leftarrow)$ Assume that there is a unique linear map $f:Uto V$ such that $fcirc g=i$, we have to show that $(U,j)$ is a free vector space on $X$.
Let $W$ be a vector space and a function $q:Xto W$, since $(V,i)$ be a free vector space on a non-empty set $X$ , by the UMP there is a unique linear map $d:Vto W$.
I just need to show that $T=fcirc d$ is unique .
I have some question , $f$ and $d$ are unique then $T$ is unique ? (i think not true) or how to show by uniqueness.
linear-algebra multilinear-algebra universal-property
linear-algebra multilinear-algebra universal-property
asked Nov 21 at 15:42
kim wenasa
1
asked Nov 21 at 15:42
kim wenasa
1
asked Nov 21 at 15:42
kim wenasa
1
asked Nov 21 at 15:42
kim wenasa
1
asked Nov 21 at 15:42
kim wenasa
1
1
You forgot to state the universal property in question. Of course it's easy to guess what it must be, but...
– David C. Ullrich
Nov 22 at 17:34
add a comment |
You forgot to state the universal property in question. Of course it's easy to guess what it must be, but...
– David C. Ullrich
Nov 22 at 17:34
You forgot to state the universal property in question. Of course it's easy to guess what it must be, but...
– David C. Ullrich
Nov 22 at 17:34
You forgot to state the universal property in question. Of course it's easy to guess what it must be, but...
– David C. Ullrich
Nov 22 at 17:34
add a comment |
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You forgot to state the universal property in question. Of course it's easy to guess what it must be, but...
– David C. Ullrich
Nov 22 at 17:34