Question on Linear Algebra (related to Affine Combination)
$begingroup$
The book I am using mentions this theorem:
$$Asubseteq Vtext{ is a flat} iff Atext{ is closed under affine combinations, that is}\ x_1,ldots,x_kin A, alpha_1,ldots,alpha_k,in F, alpha_1+cdots+ alpha_k=1Rightarrow alpha_1x_1+cdots+alpha_kx_kin A$$
The following question within the exercise then refers to this theorem (specifically the RHS of the above iff statement):
Let $ 1+1neq0$ in $F$.
Show that closure under affine combinations for $k=2$ implies closure under affine combinations for all $kgeq1$.
My approach so far has been to use Strong Induction.
Assuming the result holds for $kleq n$, we show it holds for $k=n+1$
consider $sum_{i=1}^{k+1}alpha_ix_i $
choose $lin{1,2,ldots, k+1}$ such that $sum_{i=1}^{k+1}alpha_ix_i=betasum_{i=1}^{l}frac{alpha_i}{beta}x_i+gammasum_{i={l+1}}^{k+1}frac{alpha_i}{gamma}x_i$
(Note: $beta = sum_{i=1}^{l}alpha_i neq0$ and $gamma = sum_{i=l+1}^{k+1}alpha_i neq0$)
Using induction hypothesis, $sum_{i=1}^{l}frac{alpha_i}{beta}x_iin A$ and $sum_{i={l+1}}^{k+1}frac{alpha_i}{gamma}x_iin A$. Call them $x$ and $y$ respectively.}
Now, $sum_{i=1}^{k+1}alpha_ix_i = beta x+ gamma y$
Since the result holds for $k=2$, $sum_{i=1}^{k+1}alpha_ix_iin A
$
My question is twofold. First, I never made use of the fact that $1+1neq0$ in F. What am I missing here? Second, how can I show that the condition in line 3 of my solution (when I introduce l and break the summation into two sums) can be assumed without loss of generality?
Any input would be appreciated.
linear-algebra
$endgroup$
add a comment |
$begingroup$
The book I am using mentions this theorem:
$$Asubseteq Vtext{ is a flat} iff Atext{ is closed under affine combinations, that is}\ x_1,ldots,x_kin A, alpha_1,ldots,alpha_k,in F, alpha_1+cdots+ alpha_k=1Rightarrow alpha_1x_1+cdots+alpha_kx_kin A$$
The following question within the exercise then refers to this theorem (specifically the RHS of the above iff statement):
Let $ 1+1neq0$ in $F$.
Show that closure under affine combinations for $k=2$ implies closure under affine combinations for all $kgeq1$.
My approach so far has been to use Strong Induction.
Assuming the result holds for $kleq n$, we show it holds for $k=n+1$
consider $sum_{i=1}^{k+1}alpha_ix_i $
choose $lin{1,2,ldots, k+1}$ such that $sum_{i=1}^{k+1}alpha_ix_i=betasum_{i=1}^{l}frac{alpha_i}{beta}x_i+gammasum_{i={l+1}}^{k+1}frac{alpha_i}{gamma}x_i$
(Note: $beta = sum_{i=1}^{l}alpha_i neq0$ and $gamma = sum_{i=l+1}^{k+1}alpha_i neq0$)
Using induction hypothesis, $sum_{i=1}^{l}frac{alpha_i}{beta}x_iin A$ and $sum_{i={l+1}}^{k+1}frac{alpha_i}{gamma}x_iin A$. Call them $x$ and $y$ respectively.}
Now, $sum_{i=1}^{k+1}alpha_ix_i = beta x+ gamma y$
Since the result holds for $k=2$, $sum_{i=1}^{k+1}alpha_ix_iin A
$
My question is twofold. First, I never made use of the fact that $1+1neq0$ in F. What am I missing here? Second, how can I show that the condition in line 3 of my solution (when I introduce l and break the summation into two sums) can be assumed without loss of generality?
Any input would be appreciated.
linear-algebra
$endgroup$
add a comment |
$begingroup$
The book I am using mentions this theorem:
$$Asubseteq Vtext{ is a flat} iff Atext{ is closed under affine combinations, that is}\ x_1,ldots,x_kin A, alpha_1,ldots,alpha_k,in F, alpha_1+cdots+ alpha_k=1Rightarrow alpha_1x_1+cdots+alpha_kx_kin A$$
The following question within the exercise then refers to this theorem (specifically the RHS of the above iff statement):
Let $ 1+1neq0$ in $F$.
Show that closure under affine combinations for $k=2$ implies closure under affine combinations for all $kgeq1$.
My approach so far has been to use Strong Induction.
Assuming the result holds for $kleq n$, we show it holds for $k=n+1$
consider $sum_{i=1}^{k+1}alpha_ix_i $
choose $lin{1,2,ldots, k+1}$ such that $sum_{i=1}^{k+1}alpha_ix_i=betasum_{i=1}^{l}frac{alpha_i}{beta}x_i+gammasum_{i={l+1}}^{k+1}frac{alpha_i}{gamma}x_i$
(Note: $beta = sum_{i=1}^{l}alpha_i neq0$ and $gamma = sum_{i=l+1}^{k+1}alpha_i neq0$)
Using induction hypothesis, $sum_{i=1}^{l}frac{alpha_i}{beta}x_iin A$ and $sum_{i={l+1}}^{k+1}frac{alpha_i}{gamma}x_iin A$. Call them $x$ and $y$ respectively.}
Now, $sum_{i=1}^{k+1}alpha_ix_i = beta x+ gamma y$
Since the result holds for $k=2$, $sum_{i=1}^{k+1}alpha_ix_iin A
$
My question is twofold. First, I never made use of the fact that $1+1neq0$ in F. What am I missing here? Second, how can I show that the condition in line 3 of my solution (when I introduce l and break the summation into two sums) can be assumed without loss of generality?
Any input would be appreciated.
linear-algebra
$endgroup$
The book I am using mentions this theorem:
$$Asubseteq Vtext{ is a flat} iff Atext{ is closed under affine combinations, that is}\ x_1,ldots,x_kin A, alpha_1,ldots,alpha_k,in F, alpha_1+cdots+ alpha_k=1Rightarrow alpha_1x_1+cdots+alpha_kx_kin A$$
The following question within the exercise then refers to this theorem (specifically the RHS of the above iff statement):
Let $ 1+1neq0$ in $F$.
Show that closure under affine combinations for $k=2$ implies closure under affine combinations for all $kgeq1$.
My approach so far has been to use Strong Induction.
Assuming the result holds for $kleq n$, we show it holds for $k=n+1$
consider $sum_{i=1}^{k+1}alpha_ix_i $
choose $lin{1,2,ldots, k+1}$ such that $sum_{i=1}^{k+1}alpha_ix_i=betasum_{i=1}^{l}frac{alpha_i}{beta}x_i+gammasum_{i={l+1}}^{k+1}frac{alpha_i}{gamma}x_i$
(Note: $beta = sum_{i=1}^{l}alpha_i neq0$ and $gamma = sum_{i=l+1}^{k+1}alpha_i neq0$)
Using induction hypothesis, $sum_{i=1}^{l}frac{alpha_i}{beta}x_iin A$ and $sum_{i={l+1}}^{k+1}frac{alpha_i}{gamma}x_iin A$. Call them $x$ and $y$ respectively.}
Now, $sum_{i=1}^{k+1}alpha_ix_i = beta x+ gamma y$
Since the result holds for $k=2$, $sum_{i=1}^{k+1}alpha_ix_iin A
$
My question is twofold. First, I never made use of the fact that $1+1neq0$ in F. What am I missing here? Second, how can I show that the condition in line 3 of my solution (when I introduce l and break the summation into two sums) can be assumed without loss of generality?
Any input would be appreciated.
linear-algebra
linear-algebra
edited Dec 7 '18 at 20:02
Henning Makholm
239k17304541
239k17304541
asked Dec 7 '18 at 19:07
s0ulr3aper07s0ulr3aper07
1499
1499
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
The assumption about the characteristic of $F$ comes into play when we justify the claim you ask about, so the questions are one in the same. Here's a start...
Firstly note that you can assume all the $alpha_i$ are nonzero.
Then start by dividing the sum into $beta' = alpha_n$ and $gamma' = sum_{i<n} alpha_i$. We split the problem into cases.
Case (1) $gamma' not = 0$ and there is nothing to prove.
Case (2) $gamma' = 0$ and $alpha_j not= -alpha_n$ for some $j <n$. Then we modify the division as $beta = alpha_n + a_j$ and $gamma = sum_{i not= j,n}alpha_i$ and are done.
Case (3) $gamma' = 0$ and $alpha_j = -alpha_n$ for all $j <n$. By assumption, $2alpha_j not= 0$. Hence, if we set $$gamma = gamma' + 2alpha_n = sum_{j< n-2} alpha_j$$ $$beta = beta' - 2alpha_n = sum_{igeq n-2} alpha_i = -alpha_n$$ then $beta, gamma not= 0$
$endgroup$
$begingroup$
If $gamma = 0$, then $beta = alpha_n = 1$. Going to case (3) that you've mentioned, it would imply that $gamma = 0=-alpha_n-alpha_n-cdots -alpha_n=-1-1-cdots - 1$. Now how would I proceed from this point and use $1+1neq 0$?
$endgroup$
– s0ulr3aper07
Dec 7 '18 at 22:18
1
$begingroup$
i edited a better explanation for case (3), hope that helps
$endgroup$
– Badam Baplan
Dec 7 '18 at 23:04
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%2f3030260%2fquestion-on-linear-algebra-related-to-affine-combination%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
$begingroup$
The assumption about the characteristic of $F$ comes into play when we justify the claim you ask about, so the questions are one in the same. Here's a start...
Firstly note that you can assume all the $alpha_i$ are nonzero.
Then start by dividing the sum into $beta' = alpha_n$ and $gamma' = sum_{i<n} alpha_i$. We split the problem into cases.
Case (1) $gamma' not = 0$ and there is nothing to prove.
Case (2) $gamma' = 0$ and $alpha_j not= -alpha_n$ for some $j <n$. Then we modify the division as $beta = alpha_n + a_j$ and $gamma = sum_{i not= j,n}alpha_i$ and are done.
Case (3) $gamma' = 0$ and $alpha_j = -alpha_n$ for all $j <n$. By assumption, $2alpha_j not= 0$. Hence, if we set $$gamma = gamma' + 2alpha_n = sum_{j< n-2} alpha_j$$ $$beta = beta' - 2alpha_n = sum_{igeq n-2} alpha_i = -alpha_n$$ then $beta, gamma not= 0$
$endgroup$
$begingroup$
If $gamma = 0$, then $beta = alpha_n = 1$. Going to case (3) that you've mentioned, it would imply that $gamma = 0=-alpha_n-alpha_n-cdots -alpha_n=-1-1-cdots - 1$. Now how would I proceed from this point and use $1+1neq 0$?
$endgroup$
– s0ulr3aper07
Dec 7 '18 at 22:18
1
$begingroup$
i edited a better explanation for case (3), hope that helps
$endgroup$
– Badam Baplan
Dec 7 '18 at 23:04
add a comment |
$begingroup$
The assumption about the characteristic of $F$ comes into play when we justify the claim you ask about, so the questions are one in the same. Here's a start...
Firstly note that you can assume all the $alpha_i$ are nonzero.
Then start by dividing the sum into $beta' = alpha_n$ and $gamma' = sum_{i<n} alpha_i$. We split the problem into cases.
Case (1) $gamma' not = 0$ and there is nothing to prove.
Case (2) $gamma' = 0$ and $alpha_j not= -alpha_n$ for some $j <n$. Then we modify the division as $beta = alpha_n + a_j$ and $gamma = sum_{i not= j,n}alpha_i$ and are done.
Case (3) $gamma' = 0$ and $alpha_j = -alpha_n$ for all $j <n$. By assumption, $2alpha_j not= 0$. Hence, if we set $$gamma = gamma' + 2alpha_n = sum_{j< n-2} alpha_j$$ $$beta = beta' - 2alpha_n = sum_{igeq n-2} alpha_i = -alpha_n$$ then $beta, gamma not= 0$
$endgroup$
$begingroup$
If $gamma = 0$, then $beta = alpha_n = 1$. Going to case (3) that you've mentioned, it would imply that $gamma = 0=-alpha_n-alpha_n-cdots -alpha_n=-1-1-cdots - 1$. Now how would I proceed from this point and use $1+1neq 0$?
$endgroup$
– s0ulr3aper07
Dec 7 '18 at 22:18
1
$begingroup$
i edited a better explanation for case (3), hope that helps
$endgroup$
– Badam Baplan
Dec 7 '18 at 23:04
add a comment |
$begingroup$
The assumption about the characteristic of $F$ comes into play when we justify the claim you ask about, so the questions are one in the same. Here's a start...
Firstly note that you can assume all the $alpha_i$ are nonzero.
Then start by dividing the sum into $beta' = alpha_n$ and $gamma' = sum_{i<n} alpha_i$. We split the problem into cases.
Case (1) $gamma' not = 0$ and there is nothing to prove.
Case (2) $gamma' = 0$ and $alpha_j not= -alpha_n$ for some $j <n$. Then we modify the division as $beta = alpha_n + a_j$ and $gamma = sum_{i not= j,n}alpha_i$ and are done.
Case (3) $gamma' = 0$ and $alpha_j = -alpha_n$ for all $j <n$. By assumption, $2alpha_j not= 0$. Hence, if we set $$gamma = gamma' + 2alpha_n = sum_{j< n-2} alpha_j$$ $$beta = beta' - 2alpha_n = sum_{igeq n-2} alpha_i = -alpha_n$$ then $beta, gamma not= 0$
$endgroup$
The assumption about the characteristic of $F$ comes into play when we justify the claim you ask about, so the questions are one in the same. Here's a start...
Firstly note that you can assume all the $alpha_i$ are nonzero.
Then start by dividing the sum into $beta' = alpha_n$ and $gamma' = sum_{i<n} alpha_i$. We split the problem into cases.
Case (1) $gamma' not = 0$ and there is nothing to prove.
Case (2) $gamma' = 0$ and $alpha_j not= -alpha_n$ for some $j <n$. Then we modify the division as $beta = alpha_n + a_j$ and $gamma = sum_{i not= j,n}alpha_i$ and are done.
Case (3) $gamma' = 0$ and $alpha_j = -alpha_n$ for all $j <n$. By assumption, $2alpha_j not= 0$. Hence, if we set $$gamma = gamma' + 2alpha_n = sum_{j< n-2} alpha_j$$ $$beta = beta' - 2alpha_n = sum_{igeq n-2} alpha_i = -alpha_n$$ then $beta, gamma not= 0$
edited Dec 7 '18 at 23:02
answered Dec 7 '18 at 20:01
Badam BaplanBadam Baplan
4,501722
4,501722
$begingroup$
If $gamma = 0$, then $beta = alpha_n = 1$. Going to case (3) that you've mentioned, it would imply that $gamma = 0=-alpha_n-alpha_n-cdots -alpha_n=-1-1-cdots - 1$. Now how would I proceed from this point and use $1+1neq 0$?
$endgroup$
– s0ulr3aper07
Dec 7 '18 at 22:18
1
$begingroup$
i edited a better explanation for case (3), hope that helps
$endgroup$
– Badam Baplan
Dec 7 '18 at 23:04
add a comment |
$begingroup$
If $gamma = 0$, then $beta = alpha_n = 1$. Going to case (3) that you've mentioned, it would imply that $gamma = 0=-alpha_n-alpha_n-cdots -alpha_n=-1-1-cdots - 1$. Now how would I proceed from this point and use $1+1neq 0$?
$endgroup$
– s0ulr3aper07
Dec 7 '18 at 22:18
1
$begingroup$
i edited a better explanation for case (3), hope that helps
$endgroup$
– Badam Baplan
Dec 7 '18 at 23:04
$begingroup$
If $gamma = 0$, then $beta = alpha_n = 1$. Going to case (3) that you've mentioned, it would imply that $gamma = 0=-alpha_n-alpha_n-cdots -alpha_n=-1-1-cdots - 1$. Now how would I proceed from this point and use $1+1neq 0$?
$endgroup$
– s0ulr3aper07
Dec 7 '18 at 22:18
$begingroup$
If $gamma = 0$, then $beta = alpha_n = 1$. Going to case (3) that you've mentioned, it would imply that $gamma = 0=-alpha_n-alpha_n-cdots -alpha_n=-1-1-cdots - 1$. Now how would I proceed from this point and use $1+1neq 0$?
$endgroup$
– s0ulr3aper07
Dec 7 '18 at 22:18
1
1
$begingroup$
i edited a better explanation for case (3), hope that helps
$endgroup$
– Badam Baplan
Dec 7 '18 at 23:04
$begingroup$
i edited a better explanation for case (3), hope that helps
$endgroup$
– Badam Baplan
Dec 7 '18 at 23:04
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%2f3030260%2fquestion-on-linear-algebra-related-to-affine-combination%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