Finding the dimension of $U={p(x) in mathbb{R}_3[x]text{ } |text{ } p(a) = 0}$.
$begingroup$
Let $a in mathbb{R}$ and $U = {p(x) in mathbb{R}_3[x]text{ } |text{ } p(a) = 0}$. How do I find $dim(U)$?
linear-algebra
edited Dec 23 '18 at 16:47
projectilemotion
11.4k62141
asked Dec 23 '18 at 16:38
Yuki1112Yuki1112
174
$endgroup$
add a comment |
$begingroup$
Let $a in mathbb{R}$ and $U = {p(x) in mathbb{R}_3[x]text{ } |text{ } p(a) = 0}$. How do I find $dim(U)$?
linear-algebra
edited Dec 23 '18 at 16:47
projectilemotion
11.4k62141
asked Dec 23 '18 at 16:38
Yuki1112Yuki1112
174
$endgroup$
add a comment |
$begingroup$
Let $a in mathbb{R}$ and $U = {p(x) in mathbb{R}_3[x]text{ } |text{ } p(a) = 0}$. How do I find $dim(U)$?
linear-algebra
edited Dec 23 '18 at 16:47
projectilemotion
11.4k62141
asked Dec 23 '18 at 16:38
Yuki1112Yuki1112
174
$endgroup$
Let $a in mathbb{R}$ and $U = {p(x) in mathbb{R}_3[x]text{ } |text{ } p(a) = 0}$. How do I find $dim(U)$?
linear-algebra
linear-algebra
edited Dec 23 '18 at 16:47
projectilemotion
11.4k62141
asked Dec 23 '18 at 16:38
Yuki1112Yuki1112
174
edited Dec 23 '18 at 16:47
projectilemotion
11.4k62141
asked Dec 23 '18 at 16:38
Yuki1112Yuki1112
174
edited Dec 23 '18 at 16:47
projectilemotion
11.4k62141
edited Dec 23 '18 at 16:47
projectilemotion
11.4k62141
edited Dec 23 '18 at 16:47
projectilemotion
11.4k62141
11.4k62141
asked Dec 23 '18 at 16:38
Yuki1112Yuki1112
174
asked Dec 23 '18 at 16:38
Yuki1112Yuki1112
174
asked Dec 23 '18 at 16:38
Yuki1112Yuki1112
174
174
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Let $p(x)=a_0+a_1x+a_2x^2+a_3x^3in U,a_iinBbb R$
$p(a)=0implies a_0+a_1a+a_2a^2+a_3a^3=0$
Substitute for $a_0$ in $p(x)$,
$p(x)=-(a_1a+a_2a^2+a_3a^3)+a_1x+a_2x^2+a_3x^3=a_1(x-a)+a_2(x^2-a^2)+a_3(x^3-a^3)$
which is a linear combination of $3$ linearly independent vectors of $Bbb R_3[x]$, giving the dimension as $3$.
answered Dec 23 '18 at 18:29
Shubham JohriShubham Johri
5,204718
$endgroup$
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%2f3050472%2ffinding-the-dimension-of-u-px-in-mathbbr-3x-text-text-pa%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
$begingroup$
Let $p(x)=a_0+a_1x+a_2x^2+a_3x^3in U,a_iinBbb R$
$p(a)=0implies a_0+a_1a+a_2a^2+a_3a^3=0$
Substitute for $a_0$ in $p(x)$,
$p(x)=-(a_1a+a_2a^2+a_3a^3)+a_1x+a_2x^2+a_3x^3=a_1(x-a)+a_2(x^2-a^2)+a_3(x^3-a^3)$
which is a linear combination of $3$ linearly independent vectors of $Bbb R_3[x]$, giving the dimension as $3$.
answered Dec 23 '18 at 18:29
Shubham JohriShubham Johri
5,204718
$endgroup$
add a comment |
$begingroup$
Let $p(x)=a_0+a_1x+a_2x^2+a_3x^3in U,a_iinBbb R$
$p(a)=0implies a_0+a_1a+a_2a^2+a_3a^3=0$
Substitute for $a_0$ in $p(x)$,
$p(x)=-(a_1a+a_2a^2+a_3a^3)+a_1x+a_2x^2+a_3x^3=a_1(x-a)+a_2(x^2-a^2)+a_3(x^3-a^3)$
which is a linear combination of $3$ linearly independent vectors of $Bbb R_3[x]$, giving the dimension as $3$.
answered Dec 23 '18 at 18:29
Shubham JohriShubham Johri
5,204718
$endgroup$
add a comment |
$begingroup$
Let $p(x)=a_0+a_1x+a_2x^2+a_3x^3in U,a_iinBbb R$
$p(a)=0implies a_0+a_1a+a_2a^2+a_3a^3=0$
Substitute for $a_0$ in $p(x)$,
$p(x)=-(a_1a+a_2a^2+a_3a^3)+a_1x+a_2x^2+a_3x^3=a_1(x-a)+a_2(x^2-a^2)+a_3(x^3-a^3)$
which is a linear combination of $3$ linearly independent vectors of $Bbb R_3[x]$, giving the dimension as $3$.
answered Dec 23 '18 at 18:29
Shubham JohriShubham Johri
5,204718
$endgroup$
Let $p(x)=a_0+a_1x+a_2x^2+a_3x^3in U,a_iinBbb R$
$p(a)=0implies a_0+a_1a+a_2a^2+a_3a^3=0$
Substitute for $a_0$ in $p(x)$,
$p(x)=-(a_1a+a_2a^2+a_3a^3)+a_1x+a_2x^2+a_3x^3=a_1(x-a)+a_2(x^2-a^2)+a_3(x^3-a^3)$
which is a linear combination of $3$ linearly independent vectors of $Bbb R_3[x]$, giving the dimension as $3$.
answered Dec 23 '18 at 18:29
Shubham JohriShubham Johri
5,204718
answered Dec 23 '18 at 18:29
Shubham JohriShubham Johri
5,204718
answered Dec 23 '18 at 18:29
Shubham JohriShubham Johri
5,204718
answered Dec 23 '18 at 18:29
Shubham JohriShubham Johri
5,204718
5,204718
add a comment |
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%2f3050472%2ffinding-the-dimension-of-u-px-in-mathbbr-3x-text-text-pa%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
