The Universal Mapping Property of a free vector space.
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
add a comment |
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
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
favorite
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 |
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3007891%2fthe-universal-mapping-property-of-a-free-vector-space%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3007891%2fthe-universal-mapping-property-of-a-free-vector-space%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
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