Mixing
Let $E=[1,2,…n]$ where $n$ is an odd integer.
1
$begingroup$
Let $E=[1,2,...n]$ where $n$ is an odd integer.
Let $V:$ vector space of all functions from $E$ to $mathbb{R}^3$ such that $(f+g)(k)=f(k)+g(k)$ and $(lambda f)(k)=lambda f(k)$ where $k in E$ and $f,g in V$.
We need to find the dimension of $V$.
I am confused on how to check the linear independence of the functions in $V$.
Help
linear-algebra vector-spaces linear-transformations
asked Jan 21 at 6:56
Legend KillerLegend Killer
1,6671524
$endgroup$
add a comment |
1
$begingroup$
Let $E=[1,2,...n]$ where $n$ is an odd integer.
Let $V:$ vector space of all functions from $E$ to $mathbb{R}^3$ such that $(f+g)(k)=f(k)+g(k)$ and $(lambda f)(k)=lambda f(k)$ where $k in E$ and $f,g in V$.
We need to find the dimension of $V$.
I am confused on how to check the linear independence of the functions in $V$.
Help
linear-algebra vector-spaces linear-transformations
asked Jan 21 at 6:56
Legend KillerLegend Killer
1,6671524
$endgroup$
1
$begingroup$
You check linear independence the same way you always do: is there a non-trivial linear combination which is equal to $0$? Try with $n=1$ and $n=3$ first, write down actual functions, add them together, see what you get. Then see if you can answer the questions for those cases. Then maybe you can go to the general case.
$endgroup$
– Arthur
Jan 21 at 7:15
$begingroup$
Yes, I actually got the independence case but I am not really sure about the dimension of $V$ whether it is $3$ or $3n$?
$endgroup$
– Legend Killer
Jan 21 at 12:38
$begingroup$
Well, if those arethe only two options you have to choose from: Take $n = 3$ and all shall become clear. Can you think of 4 linearly independent functions (using, say, 3 linearly independent functions you already know about from $n = 1$).
$endgroup$
– Arthur
Jan 21 at 12:47
add a comment |
1
1
1
1
$begingroup$
Let $E=[1,2,...n]$ where $n$ is an odd integer.
Let $V:$ vector space of all functions from $E$ to $mathbb{R}^3$ such that $(f+g)(k)=f(k)+g(k)$ and $(lambda f)(k)=lambda f(k)$ where $k in E$ and $f,g in V$.
We need to find the dimension of $V$.
I am confused on how to check the linear independence of the functions in $V$.
Help
linear-algebra vector-spaces linear-transformations
asked Jan 21 at 6:56
Legend KillerLegend Killer
1,6671524
$endgroup$
Let $E=[1,2,...n]$ where $n$ is an odd integer.
Let $V:$ vector space of all functions from $E$ to $mathbb{R}^3$ such that $(f+g)(k)=f(k)+g(k)$ and $(lambda f)(k)=lambda f(k)$ where $k in E$ and $f,g in V$.
We need to find the dimension of $V$.
I am confused on how to check the linear independence of the functions in $V$.
Help
linear-algebra vector-spaces linear-transformations
linear-algebra vector-spaces linear-transformations
asked Jan 21 at 6:56
Legend KillerLegend Killer
1,6671524
asked Jan 21 at 6:56
Legend KillerLegend Killer
1,6671524
asked Jan 21 at 6:56
Legend KillerLegend Killer
1,6671524
asked Jan 21 at 6:56
Legend KillerLegend Killer
1,6671524
asked Jan 21 at 6:56
Legend KillerLegend Killer
1,6671524
1,6671524
1
$begingroup$
You check linear independence the same way you always do: is there a non-trivial linear combination which is equal to $0$? Try with $n=1$ and $n=3$ first, write down actual functions, add them together, see what you get. Then see if you can answer the questions for those cases. Then maybe you can go to the general case.
$endgroup$
– Arthur
Jan 21 at 7:15
$begingroup$
Yes, I actually got the independence case but I am not really sure about the dimension of $V$ whether it is $3$ or $3n$?
$endgroup$
– Legend Killer
Jan 21 at 12:38
$begingroup$
Well, if those arethe only two options you have to choose from: Take $n = 3$ and all shall become clear. Can you think of 4 linearly independent functions (using, say, 3 linearly independent functions you already know about from $n = 1$).
$endgroup$
– Arthur
Jan 21 at 12:47
add a comment |
1
$begingroup$
You check linear independence the same way you always do: is there a non-trivial linear combination which is equal to $0$? Try with $n=1$ and $n=3$ first, write down actual functions, add them together, see what you get. Then see if you can answer the questions for those cases. Then maybe you can go to the general case.
$endgroup$
– Arthur
Jan 21 at 7:15
$begingroup$
Yes, I actually got the independence case but I am not really sure about the dimension of $V$ whether it is $3$ or $3n$?
$endgroup$
– Legend Killer
Jan 21 at 12:38
$begingroup$
Well, if those arethe only two options you have to choose from: Take $n = 3$ and all shall become clear. Can you think of 4 linearly independent functions (using, say, 3 linearly independent functions you already know about from $n = 1$).
$endgroup$
– Arthur
Jan 21 at 12:47
1
1
$begingroup$
You check linear independence the same way you always do: is there a non-trivial linear combination which is equal to $0$? Try with $n=1$ and $n=3$ first, write down actual functions, add them together, see what you get. Then see if you can answer the questions for those cases. Then maybe you can go to the general case.
$endgroup$
– Arthur
Jan 21 at 7:15
$begingroup$
You check linear independence the same way you always do: is there a non-trivial linear combination which is equal to $0$? Try with $n=1$ and $n=3$ first, write down actual functions, add them together, see what you get. Then see if you can answer the questions for those cases. Then maybe you can go to the general case.
$endgroup$
– Arthur
Jan 21 at 7:15
$begingroup$
Yes, I actually got the independence case but I am not really sure about the dimension of $V$ whether it is $3$ or $3n$?
$endgroup$
– Legend Killer
Jan 21 at 12:38
$begingroup$
Yes, I actually got the independence case but I am not really sure about the dimension of $V$ whether it is $3$ or $3n$?
$endgroup$
– Legend Killer
Jan 21 at 12:38
$begingroup$
Well, if those arethe only two options you have to choose from: Take $n = 3$ and all shall become clear. Can you think of 4 linearly independent functions (using, say, 3 linearly independent functions you already know about from $n = 1$).
$endgroup$
– Arthur
Jan 21 at 12:47
$begingroup$
Well, if those arethe only two options you have to choose from: Take $n = 3$ and all shall become clear. Can you think of 4 linearly independent functions (using, say, 3 linearly independent functions you already know about from $n = 1$).
$endgroup$
– Arthur
Jan 21 at 12:47
add a comment |
0
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
});
}
});
draft saved
draft discarded
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%2f3081581%2flet-e-1-2-n-where-n-is-an-odd-integer%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
draft saved
draft discarded
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.
draft saved
draft discarded
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%2f3081581%2flet-e-1-2-n-where-n-is-an-odd-integer%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
1
$begingroup$
You check linear independence the same way you always do: is there a non-trivial linear combination which is equal to $0$? Try with $n=1$ and $n=3$ first, write down actual functions, add them together, see what you get. Then see if you can answer the questions for those cases. Then maybe you can go to the general case.
$endgroup$
– Arthur
Jan 21 at 7:15
$begingroup$
Yes, I actually got the independence case but I am not really sure about the dimension of $V$ whether it is $3$ or $3n$?
$endgroup$
– Legend Killer
Jan 21 at 12:38
$begingroup$
Well, if those arethe only two options you have to choose from: Take $n = 3$ and all shall become clear. Can you think of 4 linearly independent functions (using, say, 3 linearly independent functions you already know about from $n = 1$).
$endgroup$
– Arthur
Jan 21 at 12:47