Basis for vector space $F(X,V)$ of functions mapping any arbitrary set to a vector space
$begingroup$
How would I go about determining the basis for the vector space $F(X,V)$ of all functions mapping an element of any arbitrary set $X$ to some vector space $V$ with the usual definition of addition and scalar multiplication on functions, should I already know the basis for that vector space?
I tried to extend the notion of the basis of the vector space of linear maps $F(X,F)$ from some set $X={a_0,...,a_n}$ to an arbitrary field where the basis was made up of functions defined as $f_i(a_j)=1$ when $i=j$ and $0$ otherwise. However I have had no such luck developing the idea for $F(X,V)$.
linear-algebra vector-spaces hamel-basis
edited Dec 10 '18 at 10:48
José Carlos Santos
158k22126228
asked Dec 8 '17 at 15:00
Michael ConnorMichael Connor
404212
$endgroup$
add a comment |
$begingroup$
How would I go about determining the basis for the vector space $F(X,V)$ of all functions mapping an element of any arbitrary set $X$ to some vector space $V$ with the usual definition of addition and scalar multiplication on functions, should I already know the basis for that vector space?
I tried to extend the notion of the basis of the vector space of linear maps $F(X,F)$ from some set $X={a_0,...,a_n}$ to an arbitrary field where the basis was made up of functions defined as $f_i(a_j)=1$ when $i=j$ and $0$ otherwise. However I have had no such luck developing the idea for $F(X,V)$.
linear-algebra vector-spaces hamel-basis
edited Dec 10 '18 at 10:48
José Carlos Santos
158k22126228
asked Dec 8 '17 at 15:00
Michael ConnorMichael Connor
404212
$endgroup$
add a comment |
$begingroup$
How would I go about determining the basis for the vector space $F(X,V)$ of all functions mapping an element of any arbitrary set $X$ to some vector space $V$ with the usual definition of addition and scalar multiplication on functions, should I already know the basis for that vector space?
I tried to extend the notion of the basis of the vector space of linear maps $F(X,F)$ from some set $X={a_0,...,a_n}$ to an arbitrary field where the basis was made up of functions defined as $f_i(a_j)=1$ when $i=j$ and $0$ otherwise. However I have had no such luck developing the idea for $F(X,V)$.
linear-algebra vector-spaces hamel-basis
edited Dec 10 '18 at 10:48
José Carlos Santos
158k22126228
asked Dec 8 '17 at 15:00
Michael ConnorMichael Connor
404212
$endgroup$
How would I go about determining the basis for the vector space $F(X,V)$ of all functions mapping an element of any arbitrary set $X$ to some vector space $V$ with the usual definition of addition and scalar multiplication on functions, should I already know the basis for that vector space?
I tried to extend the notion of the basis of the vector space of linear maps $F(X,F)$ from some set $X={a_0,...,a_n}$ to an arbitrary field where the basis was made up of functions defined as $f_i(a_j)=1$ when $i=j$ and $0$ otherwise. However I have had no such luck developing the idea for $F(X,V)$.
linear-algebra vector-spaces hamel-basis
linear-algebra vector-spaces hamel-basis
edited Dec 10 '18 at 10:48
José Carlos Santos
158k22126228
asked Dec 8 '17 at 15:00
Michael ConnorMichael Connor
404212
edited Dec 10 '18 at 10:48
José Carlos Santos
158k22126228
asked Dec 8 '17 at 15:00
Michael ConnorMichael Connor
404212
edited Dec 10 '18 at 10:48
José Carlos Santos
158k22126228
edited Dec 10 '18 at 10:48
José Carlos Santos
158k22126228
edited Dec 10 '18 at 10:48
José Carlos Santos
158k22126228
158k22126228
asked Dec 8 '17 at 15:00
Michael ConnorMichael Connor
404212
asked Dec 8 '17 at 15:00
Michael ConnorMichael Connor
404212
asked Dec 8 '17 at 15:00
Michael ConnorMichael Connor
404212
404212
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Here's an answer, assuming that $V$ is finite-dimensional. Let $B$ be a basis of $V$. For each $xin X$ and each $win B$, let $f_{x,w}in F(X,V)$ be the function defined by$$f_{x,w}(y)=begin{cases}w&text{ if }y=x\0&text{ otherwise.}end{cases}$$Then ${f_{x,w},|,xin Xwedge win B}$ is a basis of $F(X,V)$.
answered Dec 8 '17 at 15:06
José Carlos SantosJosé Carlos Santos
158k22126228
$endgroup$
$begingroup$
This does'nt work if $X$ is infinite : for example a non-zero constant function can't be obtained as a finite linear combination of $f_{x,w}$, as it would be zero for some $y$. This is also true for the case $V=F$ mentioned in the OP.
$endgroup$
– Arnaud D.
Dec 8 '17 at 15:30
add a comment |
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%2f2557254%2fbasis-for-vector-space-fx-v-of-functions-mapping-any-arbitrary-set-to-a-vect%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Here's an answer, assuming that $V$ is finite-dimensional. Let $B$ be a basis of $V$. For each $xin X$ and each $win B$, let $f_{x,w}in F(X,V)$ be the function defined by$$f_{x,w}(y)=begin{cases}w&text{ if }y=x\0&text{ otherwise.}end{cases}$$Then ${f_{x,w},|,xin Xwedge win B}$ is a basis of $F(X,V)$.
answered Dec 8 '17 at 15:06
José Carlos SantosJosé Carlos Santos
158k22126228
$endgroup$
$begingroup$
This does'nt work if $X$ is infinite : for example a non-zero constant function can't be obtained as a finite linear combination of $f_{x,w}$, as it would be zero for some $y$. This is also true for the case $V=F$ mentioned in the OP.
$endgroup$
– Arnaud D.
Dec 8 '17 at 15:30
add a comment |
$begingroup$
Here's an answer, assuming that $V$ is finite-dimensional. Let $B$ be a basis of $V$. For each $xin X$ and each $win B$, let $f_{x,w}in F(X,V)$ be the function defined by$$f_{x,w}(y)=begin{cases}w&text{ if }y=x\0&text{ otherwise.}end{cases}$$Then ${f_{x,w},|,xin Xwedge win B}$ is a basis of $F(X,V)$.
answered Dec 8 '17 at 15:06
José Carlos SantosJosé Carlos Santos
158k22126228
$endgroup$
$begingroup$
This does'nt work if $X$ is infinite : for example a non-zero constant function can't be obtained as a finite linear combination of $f_{x,w}$, as it would be zero for some $y$. This is also true for the case $V=F$ mentioned in the OP.
$endgroup$
– Arnaud D.
Dec 8 '17 at 15:30
add a comment |
$begingroup$
Here's an answer, assuming that $V$ is finite-dimensional. Let $B$ be a basis of $V$. For each $xin X$ and each $win B$, let $f_{x,w}in F(X,V)$ be the function defined by$$f_{x,w}(y)=begin{cases}w&text{ if }y=x\0&text{ otherwise.}end{cases}$$Then ${f_{x,w},|,xin Xwedge win B}$ is a basis of $F(X,V)$.
answered Dec 8 '17 at 15:06
José Carlos SantosJosé Carlos Santos
158k22126228
$endgroup$
Here's an answer, assuming that $V$ is finite-dimensional. Let $B$ be a basis of $V$. For each $xin X$ and each $win B$, let $f_{x,w}in F(X,V)$ be the function defined by$$f_{x,w}(y)=begin{cases}w&text{ if }y=x\0&text{ otherwise.}end{cases}$$Then ${f_{x,w},|,xin Xwedge win B}$ is a basis of $F(X,V)$.
answered Dec 8 '17 at 15:06
José Carlos SantosJosé Carlos Santos
158k22126228
answered Dec 8 '17 at 15:06
José Carlos SantosJosé Carlos Santos
158k22126228
answered Dec 8 '17 at 15:06
José Carlos SantosJosé Carlos Santos
158k22126228
answered Dec 8 '17 at 15:06
José Carlos SantosJosé Carlos Santos
158k22126228
158k22126228
$begingroup$
This does'nt work if $X$ is infinite : for example a non-zero constant function can't be obtained as a finite linear combination of $f_{x,w}$, as it would be zero for some $y$. This is also true for the case $V=F$ mentioned in the OP.
$endgroup$
– Arnaud D.
Dec 8 '17 at 15:30
add a comment |
$begingroup$
This does'nt work if $X$ is infinite : for example a non-zero constant function can't be obtained as a finite linear combination of $f_{x,w}$, as it would be zero for some $y$. This is also true for the case $V=F$ mentioned in the OP.
$endgroup$
– Arnaud D.
Dec 8 '17 at 15:30
$begingroup$
This does'nt work if $X$ is infinite : for example a non-zero constant function can't be obtained as a finite linear combination of $f_{x,w}$, as it would be zero for some $y$. This is also true for the case $V=F$ mentioned in the OP.
$endgroup$
– Arnaud D.
Dec 8 '17 at 15:30
$begingroup$
This does'nt work if $X$ is infinite : for example a non-zero constant function can't be obtained as a finite linear combination of $f_{x,w}$, as it would be zero for some $y$. This is also true for the case $V=F$ mentioned in the OP.
$endgroup$
– Arnaud D.
Dec 8 '17 at 15:30
add a comment |
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.
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%2f2557254%2fbasis-for-vector-space-fx-v-of-functions-mapping-any-arbitrary-set-to-a-vect%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
