I have some trouble with proof
$begingroup$
I have follow theorem:
Let X is not Empty
$$W = left{alpha _{1} x_{1} + alpha _{2} x_{2} + alpha _{3} x_{3} + ... alpha _{n} x_{n} | x _{i}in X , alpha in F right} Rightarrow W = <X>$$
Proof:
1)Lets check that M is subspace
$$a,b in W$$
$$a = left{ alpha _{1}x_{1}+ alpha _{2}x_{2} + alpha _{3}x_{3}+...+ alpha _{n}x_{n} right}
b = left{ beta _{1}x_{1}+ beta _{2}x_{2} + beta _{3}x_{3}+...+ beta _{n}x_{n} right}$$
$$a + b = left{ (alpha_{1} + beta_{1})x_{1}+ (alpha_{2} + beta_{2})x_{2} + (alpha_{3} + beta_{3})x_{3}+...+ (alpha_{n} + beta_{n})x_{n} right}in W$$
and check for multiply by the scalar
$$alpha *a = alpha _{1}alpha+ alpha _{2}alpha+ alpha _{3}alpha+...+ alpha _{n}alpha$$
2)$$Xsubseteq W (x_{i} * 1 = x_{i}) forall x in X Rightarrow xin WRightarrow x_{i} in X Rightarrow x_{i}in WRightarrow x_{i} = alpha_{1}x_{1}+ alpha_{2}x_{2}+...+ alpha_{n}x_{n}$$
where
$$alpha_{i} = 0$$
3) $$left<X right> = bigcap_{alpha}^{}{}M_{alpha} subseteq W$$
4) $$forall a = left{ alpha _{1}x_{1}+ alpha _{2}x_{2} + alpha _{3}x_{3}+...+ alpha _{n}x_{n} right} in W Rightarrow Wsubseteq left< X right>$$
What does number 3 mean? Why is this fair?
linear-algebra
asked Jan 6 at 10:09
YornYorn
85
$endgroup$
add a comment |
$begingroup$
I have follow theorem:
Let X is not Empty
$$W = left{alpha _{1} x_{1} + alpha _{2} x_{2} + alpha _{3} x_{3} + ... alpha _{n} x_{n} | x _{i}in X , alpha in F right} Rightarrow W = <X>$$
Proof:
1)Lets check that M is subspace
$$a,b in W$$
$$a = left{ alpha _{1}x_{1}+ alpha _{2}x_{2} + alpha _{3}x_{3}+...+ alpha _{n}x_{n} right}
b = left{ beta _{1}x_{1}+ beta _{2}x_{2} + beta _{3}x_{3}+...+ beta _{n}x_{n} right}$$
$$a + b = left{ (alpha_{1} + beta_{1})x_{1}+ (alpha_{2} + beta_{2})x_{2} + (alpha_{3} + beta_{3})x_{3}+...+ (alpha_{n} + beta_{n})x_{n} right}in W$$
and check for multiply by the scalar
$$alpha *a = alpha _{1}alpha+ alpha _{2}alpha+ alpha _{3}alpha+...+ alpha _{n}alpha$$
2)$$Xsubseteq W (x_{i} * 1 = x_{i}) forall x in X Rightarrow xin WRightarrow x_{i} in X Rightarrow x_{i}in WRightarrow x_{i} = alpha_{1}x_{1}+ alpha_{2}x_{2}+...+ alpha_{n}x_{n}$$
where
$$alpha_{i} = 0$$
3) $$left<X right> = bigcap_{alpha}^{}{}M_{alpha} subseteq W$$
4) $$forall a = left{ alpha _{1}x_{1}+ alpha _{2}x_{2} + alpha _{3}x_{3}+...+ alpha _{n}x_{n} right} in W Rightarrow Wsubseteq left< X right>$$
What does number 3 mean? Why is this fair?
linear-algebra
asked Jan 6 at 10:09
YornYorn
85
$endgroup$
$begingroup$
Could you first define $M_{alpha}$ in 3)?
$endgroup$
– Dietrich Burde
Jan 6 at 10:12
$begingroup$
$$M_{alpha}$$ It's the set of Subspaces that contain X
$endgroup$
– Yorn
Jan 6 at 10:49
$begingroup$
Then 3) is clear, right?
$endgroup$
– Dietrich Burde
Jan 6 at 11:03
$begingroup$
$$X = left{l_{1},l_{2},l_{3} right}$$ $$M_{1}=left{l_{1},l_{2},l_{3},v_{1} right}$$ $$M_{2}=left{l_{1},l_{2},l_{3},v_{1},v_{2} right}$$ $$M_{3}=left{l_{1},l_{2},l_{3},v_{1},v_{2},v_{3} right}$$ $$bigcap_{alpha = 1}^{3}{}M_{alpha} =left{l_{1},l_{2},l_{3},v_{1} right}$$ $$so,: why: can: we: say: that: v_{1}: in: W?$$
$endgroup$
– Yorn
Jan 6 at 12:11
$begingroup$
It is the intersection of all subspaces containing $X$, not just of $3$ subspaces. You need to review the definition of $cap_{alpha}M_{alpha}$.
$endgroup$
– Dietrich Burde
Jan 6 at 12:30
add a comment |
$begingroup$
I have follow theorem:
Let X is not Empty
$$W = left{alpha _{1} x_{1} + alpha _{2} x_{2} + alpha _{3} x_{3} + ... alpha _{n} x_{n} | x _{i}in X , alpha in F right} Rightarrow W = <X>$$
Proof:
1)Lets check that M is subspace
$$a,b in W$$
$$a = left{ alpha _{1}x_{1}+ alpha _{2}x_{2} + alpha _{3}x_{3}+...+ alpha _{n}x_{n} right}
b = left{ beta _{1}x_{1}+ beta _{2}x_{2} + beta _{3}x_{3}+...+ beta _{n}x_{n} right}$$
$$a + b = left{ (alpha_{1} + beta_{1})x_{1}+ (alpha_{2} + beta_{2})x_{2} + (alpha_{3} + beta_{3})x_{3}+...+ (alpha_{n} + beta_{n})x_{n} right}in W$$
and check for multiply by the scalar
$$alpha *a = alpha _{1}alpha+ alpha _{2}alpha+ alpha _{3}alpha+...+ alpha _{n}alpha$$
2)$$Xsubseteq W (x_{i} * 1 = x_{i}) forall x in X Rightarrow xin WRightarrow x_{i} in X Rightarrow x_{i}in WRightarrow x_{i} = alpha_{1}x_{1}+ alpha_{2}x_{2}+...+ alpha_{n}x_{n}$$
where
$$alpha_{i} = 0$$
3) $$left<X right> = bigcap_{alpha}^{}{}M_{alpha} subseteq W$$
4) $$forall a = left{ alpha _{1}x_{1}+ alpha _{2}x_{2} + alpha _{3}x_{3}+...+ alpha _{n}x_{n} right} in W Rightarrow Wsubseteq left< X right>$$
What does number 3 mean? Why is this fair?
linear-algebra
asked Jan 6 at 10:09
YornYorn
85
$endgroup$
I have follow theorem:
Let X is not Empty
$$W = left{alpha _{1} x_{1} + alpha _{2} x_{2} + alpha _{3} x_{3} + ... alpha _{n} x_{n} | x _{i}in X , alpha in F right} Rightarrow W = <X>$$
Proof:
1)Lets check that M is subspace
$$a,b in W$$
$$a = left{ alpha _{1}x_{1}+ alpha _{2}x_{2} + alpha _{3}x_{3}+...+ alpha _{n}x_{n} right}
b = left{ beta _{1}x_{1}+ beta _{2}x_{2} + beta _{3}x_{3}+...+ beta _{n}x_{n} right}$$
$$a + b = left{ (alpha_{1} + beta_{1})x_{1}+ (alpha_{2} + beta_{2})x_{2} + (alpha_{3} + beta_{3})x_{3}+...+ (alpha_{n} + beta_{n})x_{n} right}in W$$
and check for multiply by the scalar
$$alpha *a = alpha _{1}alpha+ alpha _{2}alpha+ alpha _{3}alpha+...+ alpha _{n}alpha$$
2)$$Xsubseteq W (x_{i} * 1 = x_{i}) forall x in X Rightarrow xin WRightarrow x_{i} in X Rightarrow x_{i}in WRightarrow x_{i} = alpha_{1}x_{1}+ alpha_{2}x_{2}+...+ alpha_{n}x_{n}$$
where
$$alpha_{i} = 0$$
3) $$left<X right> = bigcap_{alpha}^{}{}M_{alpha} subseteq W$$
4) $$forall a = left{ alpha _{1}x_{1}+ alpha _{2}x_{2} + alpha _{3}x_{3}+...+ alpha _{n}x_{n} right} in W Rightarrow Wsubseteq left< X right>$$
What does number 3 mean? Why is this fair?
linear-algebra
linear-algebra
asked Jan 6 at 10:09
YornYorn
85
asked Jan 6 at 10:09
YornYorn
85
asked Jan 6 at 10:09
YornYorn
85
asked Jan 6 at 10:09
YornYorn
85
asked Jan 6 at 10:09
YornYorn
85
85
$begingroup$
Could you first define $M_{alpha}$ in 3)?
$endgroup$
– Dietrich Burde
Jan 6 at 10:12
$begingroup$
$$M_{alpha}$$ It's the set of Subspaces that contain X
$endgroup$
– Yorn
Jan 6 at 10:49
$begingroup$
Then 3) is clear, right?
$endgroup$
– Dietrich Burde
Jan 6 at 11:03
$begingroup$
$$X = left{l_{1},l_{2},l_{3} right}$$ $$M_{1}=left{l_{1},l_{2},l_{3},v_{1} right}$$ $$M_{2}=left{l_{1},l_{2},l_{3},v_{1},v_{2} right}$$ $$M_{3}=left{l_{1},l_{2},l_{3},v_{1},v_{2},v_{3} right}$$ $$bigcap_{alpha = 1}^{3}{}M_{alpha} =left{l_{1},l_{2},l_{3},v_{1} right}$$ $$so,: why: can: we: say: that: v_{1}: in: W?$$
$endgroup$
– Yorn
Jan 6 at 12:11
$begingroup$
It is the intersection of all subspaces containing $X$, not just of $3$ subspaces. You need to review the definition of $cap_{alpha}M_{alpha}$.
$endgroup$
– Dietrich Burde
Jan 6 at 12:30
add a comment |
$begingroup$
Could you first define $M_{alpha}$ in 3)?
$endgroup$
– Dietrich Burde
Jan 6 at 10:12
$begingroup$
$$M_{alpha}$$ It's the set of Subspaces that contain X
$endgroup$
– Yorn
Jan 6 at 10:49
$begingroup$
Then 3) is clear, right?
$endgroup$
– Dietrich Burde
Jan 6 at 11:03
$begingroup$
$$X = left{l_{1},l_{2},l_{3} right}$$ $$M_{1}=left{l_{1},l_{2},l_{3},v_{1} right}$$ $$M_{2}=left{l_{1},l_{2},l_{3},v_{1},v_{2} right}$$ $$M_{3}=left{l_{1},l_{2},l_{3},v_{1},v_{2},v_{3} right}$$ $$bigcap_{alpha = 1}^{3}{}M_{alpha} =left{l_{1},l_{2},l_{3},v_{1} right}$$ $$so,: why: can: we: say: that: v_{1}: in: W?$$
$endgroup$
– Yorn
Jan 6 at 12:11
$begingroup$
It is the intersection of all subspaces containing $X$, not just of $3$ subspaces. You need to review the definition of $cap_{alpha}M_{alpha}$.
$endgroup$
– Dietrich Burde
Jan 6 at 12:30
$begingroup$
Could you first define $M_{alpha}$ in 3)?
$endgroup$
– Dietrich Burde
Jan 6 at 10:12
$begingroup$
Could you first define $M_{alpha}$ in 3)?
$endgroup$
– Dietrich Burde
Jan 6 at 10:12
$begingroup$
$$M_{alpha}$$ It's the set of Subspaces that contain X
$endgroup$
– Yorn
Jan 6 at 10:49
$begingroup$
$$M_{alpha}$$ It's the set of Subspaces that contain X
$endgroup$
– Yorn
Jan 6 at 10:49
$begingroup$
Then 3) is clear, right?
$endgroup$
– Dietrich Burde
Jan 6 at 11:03
$begingroup$
Then 3) is clear, right?
$endgroup$
– Dietrich Burde
Jan 6 at 11:03
$begingroup$
$$X = left{l_{1},l_{2},l_{3} right}$$ $$M_{1}=left{l_{1},l_{2},l_{3},v_{1} right}$$ $$M_{2}=left{l_{1},l_{2},l_{3},v_{1},v_{2} right}$$ $$M_{3}=left{l_{1},l_{2},l_{3},v_{1},v_{2},v_{3} right}$$ $$bigcap_{alpha = 1}^{3}{}M_{alpha} =left{l_{1},l_{2},l_{3},v_{1} right}$$ $$so,: why: can: we: say: that: v_{1}: in: W?$$
$endgroup$
– Yorn
Jan 6 at 12:11
$begingroup$
$$X = left{l_{1},l_{2},l_{3} right}$$ $$M_{1}=left{l_{1},l_{2},l_{3},v_{1} right}$$ $$M_{2}=left{l_{1},l_{2},l_{3},v_{1},v_{2} right}$$ $$M_{3}=left{l_{1},l_{2},l_{3},v_{1},v_{2},v_{3} right}$$ $$bigcap_{alpha = 1}^{3}{}M_{alpha} =left{l_{1},l_{2},l_{3},v_{1} right}$$ $$so,: why: can: we: say: that: v_{1}: in: W?$$
$endgroup$
– Yorn
Jan 6 at 12:11
$begingroup$
It is the intersection of all subspaces containing $X$, not just of $3$ subspaces. You need to review the definition of $cap_{alpha}M_{alpha}$.
$endgroup$
– Dietrich Burde
Jan 6 at 12:30
$begingroup$
It is the intersection of all subspaces containing $X$, not just of $3$ subspaces. You need to review the definition of $cap_{alpha}M_{alpha}$.
$endgroup$
– Dietrich Burde
Jan 6 at 12:30
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
});
}
});
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%2f3063677%2fi-have-some-trouble-with-proof%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
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%2f3063677%2fi-have-some-trouble-with-proof%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
$begingroup$
Could you first define $M_{alpha}$ in 3)?
$endgroup$
– Dietrich Burde
Jan 6 at 10:12
$begingroup$
$$M_{alpha}$$ It's the set of Subspaces that contain X
$endgroup$
– Yorn
Jan 6 at 10:49
$begingroup$
Then 3) is clear, right?
$endgroup$
– Dietrich Burde
Jan 6 at 11:03
$begingroup$
$$X = left{l_{1},l_{2},l_{3} right}$$ $$M_{1}=left{l_{1},l_{2},l_{3},v_{1} right}$$ $$M_{2}=left{l_{1},l_{2},l_{3},v_{1},v_{2} right}$$ $$M_{3}=left{l_{1},l_{2},l_{3},v_{1},v_{2},v_{3} right}$$ $$bigcap_{alpha = 1}^{3}{}M_{alpha} =left{l_{1},l_{2},l_{3},v_{1} right}$$ $$so,: why: can: we: say: that: v_{1}: in: W?$$
$endgroup$
– Yorn
Jan 6 at 12:11
$begingroup$
It is the intersection of all subspaces containing $X$, not just of $3$ subspaces. You need to review the definition of $cap_{alpha}M_{alpha}$.
$endgroup$
– Dietrich Burde
Jan 6 at 12:30