Prove that the image of the bilinear application is not a vector space
$begingroup$
Prove that the image of the bilinear application
$$
mathbb{R}^2timesmathbb{R}^2
ni big((x_1,x_2),(y_1,y_2)big)
mapsto
varphi big((x_1,x_2),(y_1,y_2)big) =
(x_1y_1,x_1y_2,x_2y_1,x_2y_2)inmathbb{R}^4
$$ is not a vector space.
My work. If $mathrm{Im}(varphi)subseteqmathbb{R}^4$ is a subspace of $ mathbb{R}^{4}$ then for any scalars $mu,lambdainmathbb{R}$ and vectors
$$
p=(acdot c, acdot d, bcdot c, bcdot d)
qquad mbox{ and } qquad
q=(ucdot z, u cdot w, vcdot z, vcdot w)
$$ in $mathrm{Im}(varphi)$ we have
$$
mucdot p+lambdacdot q in mathrm{Im}(varphi).
$$
Therefore, if we can show that there are scalars $lambda,muinmathbb{R}$ and vectors $p,qin mathrm{Im}(varphi)$ such that
$$
mucdot p+lambdacdot q notin mathrm{Im}(varphi)
$$
then $mathrm{Im}(varphi)$ will not be a vector space. The relationship of non-pertinence above is equivalent to the following statement. For all vector $(x_1y_2,x_1y_2,x_2y_2,x_2y_2)inmathrm{Im}varphi$ the system below has no solution
$$
begin{array}{rcl}
mu a c+ lambda u z &=& x_1y_1\
mu a d+ lambda u w &=& x_1y_2\
mu b c+ lambda v z &=& x_2y_1\
mu b d+ lambda v w &=& x_2y_2\
end{array}
$$
But I have been unable to find real numbers $a$, $b$, $c$ and $d$ and scalars $lambda$ and $mu$ for which the above system has no solution for $x_1$, $x_2$, $y_1$ and $y_2$ varying in $mathbb{R} $.
multilinear-algebra
asked Dec 29 '18 at 20:28
MathOverviewMathOverview
8,95043164
$endgroup$
add a comment |
$begingroup$
Prove that the image of the bilinear application
$$
mathbb{R}^2timesmathbb{R}^2
ni big((x_1,x_2),(y_1,y_2)big)
mapsto
varphi big((x_1,x_2),(y_1,y_2)big) =
(x_1y_1,x_1y_2,x_2y_1,x_2y_2)inmathbb{R}^4
$$ is not a vector space.
My work. If $mathrm{Im}(varphi)subseteqmathbb{R}^4$ is a subspace of $ mathbb{R}^{4}$ then for any scalars $mu,lambdainmathbb{R}$ and vectors
$$
p=(acdot c, acdot d, bcdot c, bcdot d)
qquad mbox{ and } qquad
q=(ucdot z, u cdot w, vcdot z, vcdot w)
$$ in $mathrm{Im}(varphi)$ we have
$$
mucdot p+lambdacdot q in mathrm{Im}(varphi).
$$
Therefore, if we can show that there are scalars $lambda,muinmathbb{R}$ and vectors $p,qin mathrm{Im}(varphi)$ such that
$$
mucdot p+lambdacdot q notin mathrm{Im}(varphi)
$$
then $mathrm{Im}(varphi)$ will not be a vector space. The relationship of non-pertinence above is equivalent to the following statement. For all vector $(x_1y_2,x_1y_2,x_2y_2,x_2y_2)inmathrm{Im}varphi$ the system below has no solution
$$
begin{array}{rcl}
mu a c+ lambda u z &=& x_1y_1\
mu a d+ lambda u w &=& x_1y_2\
mu b c+ lambda v z &=& x_2y_1\
mu b d+ lambda v w &=& x_2y_2\
end{array}
$$
But I have been unable to find real numbers $a$, $b$, $c$ and $d$ and scalars $lambda$ and $mu$ for which the above system has no solution for $x_1$, $x_2$, $y_1$ and $y_2$ varying in $mathbb{R} $.
multilinear-algebra
asked Dec 29 '18 at 20:28
MathOverviewMathOverview
8,95043164
$endgroup$
add a comment |
$begingroup$
Prove that the image of the bilinear application
$$
mathbb{R}^2timesmathbb{R}^2
ni big((x_1,x_2),(y_1,y_2)big)
mapsto
varphi big((x_1,x_2),(y_1,y_2)big) =
(x_1y_1,x_1y_2,x_2y_1,x_2y_2)inmathbb{R}^4
$$ is not a vector space.
My work. If $mathrm{Im}(varphi)subseteqmathbb{R}^4$ is a subspace of $ mathbb{R}^{4}$ then for any scalars $mu,lambdainmathbb{R}$ and vectors
$$
p=(acdot c, acdot d, bcdot c, bcdot d)
qquad mbox{ and } qquad
q=(ucdot z, u cdot w, vcdot z, vcdot w)
$$ in $mathrm{Im}(varphi)$ we have
$$
mucdot p+lambdacdot q in mathrm{Im}(varphi).
$$
Therefore, if we can show that there are scalars $lambda,muinmathbb{R}$ and vectors $p,qin mathrm{Im}(varphi)$ such that
$$
mucdot p+lambdacdot q notin mathrm{Im}(varphi)
$$
then $mathrm{Im}(varphi)$ will not be a vector space. The relationship of non-pertinence above is equivalent to the following statement. For all vector $(x_1y_2,x_1y_2,x_2y_2,x_2y_2)inmathrm{Im}varphi$ the system below has no solution
$$
begin{array}{rcl}
mu a c+ lambda u z &=& x_1y_1\
mu a d+ lambda u w &=& x_1y_2\
mu b c+ lambda v z &=& x_2y_1\
mu b d+ lambda v w &=& x_2y_2\
end{array}
$$
But I have been unable to find real numbers $a$, $b$, $c$ and $d$ and scalars $lambda$ and $mu$ for which the above system has no solution for $x_1$, $x_2$, $y_1$ and $y_2$ varying in $mathbb{R} $.
multilinear-algebra
asked Dec 29 '18 at 20:28
MathOverviewMathOverview
8,95043164
$endgroup$
Prove that the image of the bilinear application
$$
mathbb{R}^2timesmathbb{R}^2
ni big((x_1,x_2),(y_1,y_2)big)
mapsto
varphi big((x_1,x_2),(y_1,y_2)big) =
(x_1y_1,x_1y_2,x_2y_1,x_2y_2)inmathbb{R}^4
$$ is not a vector space.
My work. If $mathrm{Im}(varphi)subseteqmathbb{R}^4$ is a subspace of $ mathbb{R}^{4}$ then for any scalars $mu,lambdainmathbb{R}$ and vectors
$$
p=(acdot c, acdot d, bcdot c, bcdot d)
qquad mbox{ and } qquad
q=(ucdot z, u cdot w, vcdot z, vcdot w)
$$ in $mathrm{Im}(varphi)$ we have
$$
mucdot p+lambdacdot q in mathrm{Im}(varphi).
$$
Therefore, if we can show that there are scalars $lambda,muinmathbb{R}$ and vectors $p,qin mathrm{Im}(varphi)$ such that
$$
mucdot p+lambdacdot q notin mathrm{Im}(varphi)
$$
then $mathrm{Im}(varphi)$ will not be a vector space. The relationship of non-pertinence above is equivalent to the following statement. For all vector $(x_1y_2,x_1y_2,x_2y_2,x_2y_2)inmathrm{Im}varphi$ the system below has no solution
$$
begin{array}{rcl}
mu a c+ lambda u z &=& x_1y_1\
mu a d+ lambda u w &=& x_1y_2\
mu b c+ lambda v z &=& x_2y_1\
mu b d+ lambda v w &=& x_2y_2\
end{array}
$$
But I have been unable to find real numbers $a$, $b$, $c$ and $d$ and scalars $lambda$ and $mu$ for which the above system has no solution for $x_1$, $x_2$, $y_1$ and $y_2$ varying in $mathbb{R} $.
multilinear-algebra
multilinear-algebra
asked Dec 29 '18 at 20:28
MathOverviewMathOverview
8,95043164
asked Dec 29 '18 at 20:28
MathOverviewMathOverview
8,95043164
asked Dec 29 '18 at 20:28
MathOverviewMathOverview
8,95043164
asked Dec 29 '18 at 20:28
MathOverviewMathOverview
8,95043164
asked Dec 29 '18 at 20:28
MathOverviewMathOverview
8,95043164
8,95043164
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Your claim is true as $,operatorname{Im}varphi,$ is $,$ not closed $,$ under addition:
$$varphibig((1,0),(1,0)big) :=: (1,0,0,0)\
varphibig((0,1),(0,1)big) :=: (0,0,0,1)$$
Now consider the sum $(1,0,0,1)$ of both ...
Can it be contained in $operatorname{Im}varphi$?
answered Dec 29 '18 at 22:22
HannoHanno
2,469628
$endgroup$
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%2f3056225%2fprove-that-the-image-of-the-bilinear-application-is-not-a-vector-space%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
$begingroup$
Your claim is true as $,operatorname{Im}varphi,$ is $,$ not closed $,$ under addition:
$$varphibig((1,0),(1,0)big) :=: (1,0,0,0)\
varphibig((0,1),(0,1)big) :=: (0,0,0,1)$$
Now consider the sum $(1,0,0,1)$ of both ...
Can it be contained in $operatorname{Im}varphi$?
answered Dec 29 '18 at 22:22
HannoHanno
2,469628
$endgroup$
add a comment |
$begingroup$
Your claim is true as $,operatorname{Im}varphi,$ is $,$ not closed $,$ under addition:
$$varphibig((1,0),(1,0)big) :=: (1,0,0,0)\
varphibig((0,1),(0,1)big) :=: (0,0,0,1)$$
Now consider the sum $(1,0,0,1)$ of both ...
Can it be contained in $operatorname{Im}varphi$?
answered Dec 29 '18 at 22:22
HannoHanno
2,469628
$endgroup$
add a comment |
$begingroup$
Your claim is true as $,operatorname{Im}varphi,$ is $,$ not closed $,$ under addition:
$$varphibig((1,0),(1,0)big) :=: (1,0,0,0)\
varphibig((0,1),(0,1)big) :=: (0,0,0,1)$$
Now consider the sum $(1,0,0,1)$ of both ...
Can it be contained in $operatorname{Im}varphi$?
answered Dec 29 '18 at 22:22
HannoHanno
2,469628
$endgroup$
Your claim is true as $,operatorname{Im}varphi,$ is $,$ not closed $,$ under addition:
$$varphibig((1,0),(1,0)big) :=: (1,0,0,0)\
varphibig((0,1),(0,1)big) :=: (0,0,0,1)$$
Now consider the sum $(1,0,0,1)$ of both ...
Can it be contained in $operatorname{Im}varphi$?
answered Dec 29 '18 at 22:22
HannoHanno
2,469628
answered Dec 29 '18 at 22:22
HannoHanno
2,469628
answered Dec 29 '18 at 22:22
HannoHanno
2,469628
answered Dec 29 '18 at 22:22
HannoHanno
2,469628
2,469628
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%2f3056225%2fprove-that-the-image-of-the-bilinear-application-is-not-a-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
