Show that if ${vec{v_1},vec{v_2},vec{v_3}}$ is a linearly independent set, then ${vec{v_1},vec{v_2}}$ is also...
$begingroup$
How do you show that if ${vec{v_1},vec{v_2},vec{v_3}}$ is a linearly independent set, then ${vec{v_1},vec{v_2}}$ is also linearly independent.
I need to explain this in complete sentences. I understand how to show linear independence for specific equations but I don't have a high enough level of understand of linear independence to be able to explain this more broadly.
linear-algebra
edited Jan 21 at 22:40
idriskameni
685319
asked Jan 21 at 22:25
hoya2021hoya2021
265
$endgroup$
add a comment |
$begingroup$
How do you show that if ${vec{v_1},vec{v_2},vec{v_3}}$ is a linearly independent set, then ${vec{v_1},vec{v_2}}$ is also linearly independent.
I need to explain this in complete sentences. I understand how to show linear independence for specific equations but I don't have a high enough level of understand of linear independence to be able to explain this more broadly.
linear-algebra
edited Jan 21 at 22:40
idriskameni
685319
asked Jan 21 at 22:25
hoya2021hoya2021
265
$endgroup$
3
$begingroup$
Can you state the definitions of "${v_1, v_2, v_3}$ is a linearly independent set" and "${v_1, v_2}$ is a linearly independent set"? If you write out the definitions clearly, you should be able to finish the question in one step.
$endgroup$
– angryavian
Jan 21 at 22:28
add a comment |
$begingroup$
How do you show that if ${vec{v_1},vec{v_2},vec{v_3}}$ is a linearly independent set, then ${vec{v_1},vec{v_2}}$ is also linearly independent.
I need to explain this in complete sentences. I understand how to show linear independence for specific equations but I don't have a high enough level of understand of linear independence to be able to explain this more broadly.
linear-algebra
edited Jan 21 at 22:40
idriskameni
685319
asked Jan 21 at 22:25
hoya2021hoya2021
265
$endgroup$
How do you show that if ${vec{v_1},vec{v_2},vec{v_3}}$ is a linearly independent set, then ${vec{v_1},vec{v_2}}$ is also linearly independent.
I need to explain this in complete sentences. I understand how to show linear independence for specific equations but I don't have a high enough level of understand of linear independence to be able to explain this more broadly.
linear-algebra
linear-algebra
edited Jan 21 at 22:40
idriskameni
685319
asked Jan 21 at 22:25
hoya2021hoya2021
265
edited Jan 21 at 22:40
idriskameni
685319
asked Jan 21 at 22:25
hoya2021hoya2021
265
edited Jan 21 at 22:40
idriskameni
685319
edited Jan 21 at 22:40
idriskameni
685319
edited Jan 21 at 22:40
idriskameni
685319
685319
asked Jan 21 at 22:25
hoya2021hoya2021
265
asked Jan 21 at 22:25
hoya2021hoya2021
265
asked Jan 21 at 22:25
hoya2021hoya2021
265
265
3
$begingroup$
Can you state the definitions of "${v_1, v_2, v_3}$ is a linearly independent set" and "${v_1, v_2}$ is a linearly independent set"? If you write out the definitions clearly, you should be able to finish the question in one step.
$endgroup$
– angryavian
Jan 21 at 22:28
add a comment |
3
$begingroup$
Can you state the definitions of "${v_1, v_2, v_3}$ is a linearly independent set" and "${v_1, v_2}$ is a linearly independent set"? If you write out the definitions clearly, you should be able to finish the question in one step.
$endgroup$
– angryavian
Jan 21 at 22:28
3
3
$begingroup$
Can you state the definitions of "${v_1, v_2, v_3}$ is a linearly independent set" and "${v_1, v_2}$ is a linearly independent set"? If you write out the definitions clearly, you should be able to finish the question in one step.
$endgroup$
– angryavian
Jan 21 at 22:28
$begingroup$
Can you state the definitions of "${v_1, v_2, v_3}$ is a linearly independent set" and "${v_1, v_2}$ is a linearly independent set"? If you write out the definitions clearly, you should be able to finish the question in one step.
$endgroup$
– angryavian
Jan 21 at 22:28
add a comment |
3 Answers
3
active
oldest
votes
$begingroup$
Suppose that ${v_1,v_2}$ is a linearly dependent set. That means by definition that...
...there is some set of constants $c_1,c_2$ where at least one of $c_1,c_2$ is nonzero such that $c_1v_1 + c_2v_2 = 0$.
Then when considering whether or not ${v_1,v_2,v_3}$ is a linearly independent set or not we see that...
...by using the same $c_1,c_2$ as before and letting $c_3=0$ we have shown that there is an example of $c_1,c_2,c_3$ such that $c_1v_1+c_2v_2+c_3v_3=0$ while at least one of $c_1,c_2,c_3$ is nonzero.
So we have shown that ${v_1,v_2}$ linearly dependent implies that ${v_1,v_2,v_3}$ is linearly dependent as well. By contraposition, this is equivalent to have shown that ${v_1,v_2,v_3}$ being linearly independent implies ${v_1,v_2}$ is linearly independent as well.
answered Jan 21 at 22:38
JMoravitzJMoravitz
47.9k33886
$endgroup$
add a comment |
$begingroup$
IIRC,
if $v_{1,2,3}$
are linearly independent then
$sum a_iv_i = 0
implies
a_{1,2,3} = 0
$.
If
$v_1, v_2$
are not linearly independent,
there exist $a, b$
with at least one of
$a, b$ nonzero such that
$av_1+bv_2 =0$.
Therefore
$0
=av_1+bv_2 +0v_3$
so that
$v_{1, 2, 3}$
are not linearly independent.
Contradiction with assumption.
answered Jan 21 at 22:37
marty cohenmarty cohen
74k549128
$endgroup$
add a comment |
$begingroup$
Here's a short direct proof, without unnecessary negation:
(equivalent to the other answers)
Suppose
$a_1 v_1 + a_2 v_2 =0$.
We need to show $a_1=a_2=0$.
Since $0v_3=0$, we also have
$a_1 v_1 + a_2 v_2 + 0 v_3 =0$.
Since $v_1,v_2,v_3$ are linearly independent, $a_1=a_2=0$.
answered Jan 29 at 20:14
ffffforallffffforall
36028
$endgroup$
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%2f3082503%2fshow-that-if-vecv-1-vecv-2-vecv-3-is-a-linearly-independent-set%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Suppose that ${v_1,v_2}$ is a linearly dependent set. That means by definition that...
...there is some set of constants $c_1,c_2$ where at least one of $c_1,c_2$ is nonzero such that $c_1v_1 + c_2v_2 = 0$.
Then when considering whether or not ${v_1,v_2,v_3}$ is a linearly independent set or not we see that...
...by using the same $c_1,c_2$ as before and letting $c_3=0$ we have shown that there is an example of $c_1,c_2,c_3$ such that $c_1v_1+c_2v_2+c_3v_3=0$ while at least one of $c_1,c_2,c_3$ is nonzero.
So we have shown that ${v_1,v_2}$ linearly dependent implies that ${v_1,v_2,v_3}$ is linearly dependent as well. By contraposition, this is equivalent to have shown that ${v_1,v_2,v_3}$ being linearly independent implies ${v_1,v_2}$ is linearly independent as well.
answered Jan 21 at 22:38
JMoravitzJMoravitz
47.9k33886
$endgroup$
add a comment |
$begingroup$
Suppose that ${v_1,v_2}$ is a linearly dependent set. That means by definition that...
...there is some set of constants $c_1,c_2$ where at least one of $c_1,c_2$ is nonzero such that $c_1v_1 + c_2v_2 = 0$.
Then when considering whether or not ${v_1,v_2,v_3}$ is a linearly independent set or not we see that...
...by using the same $c_1,c_2$ as before and letting $c_3=0$ we have shown that there is an example of $c_1,c_2,c_3$ such that $c_1v_1+c_2v_2+c_3v_3=0$ while at least one of $c_1,c_2,c_3$ is nonzero.
So we have shown that ${v_1,v_2}$ linearly dependent implies that ${v_1,v_2,v_3}$ is linearly dependent as well. By contraposition, this is equivalent to have shown that ${v_1,v_2,v_3}$ being linearly independent implies ${v_1,v_2}$ is linearly independent as well.
answered Jan 21 at 22:38
JMoravitzJMoravitz
47.9k33886
$endgroup$
add a comment |
$begingroup$
Suppose that ${v_1,v_2}$ is a linearly dependent set. That means by definition that...
...there is some set of constants $c_1,c_2$ where at least one of $c_1,c_2$ is nonzero such that $c_1v_1 + c_2v_2 = 0$.
Then when considering whether or not ${v_1,v_2,v_3}$ is a linearly independent set or not we see that...
...by using the same $c_1,c_2$ as before and letting $c_3=0$ we have shown that there is an example of $c_1,c_2,c_3$ such that $c_1v_1+c_2v_2+c_3v_3=0$ while at least one of $c_1,c_2,c_3$ is nonzero.
So we have shown that ${v_1,v_2}$ linearly dependent implies that ${v_1,v_2,v_3}$ is linearly dependent as well. By contraposition, this is equivalent to have shown that ${v_1,v_2,v_3}$ being linearly independent implies ${v_1,v_2}$ is linearly independent as well.
answered Jan 21 at 22:38
JMoravitzJMoravitz
47.9k33886
$endgroup$
Suppose that ${v_1,v_2}$ is a linearly dependent set. That means by definition that...
...there is some set of constants $c_1,c_2$ where at least one of $c_1,c_2$ is nonzero such that $c_1v_1 + c_2v_2 = 0$.
Then when considering whether or not ${v_1,v_2,v_3}$ is a linearly independent set or not we see that...
...by using the same $c_1,c_2$ as before and letting $c_3=0$ we have shown that there is an example of $c_1,c_2,c_3$ such that $c_1v_1+c_2v_2+c_3v_3=0$ while at least one of $c_1,c_2,c_3$ is nonzero.
So we have shown that ${v_1,v_2}$ linearly dependent implies that ${v_1,v_2,v_3}$ is linearly dependent as well. By contraposition, this is equivalent to have shown that ${v_1,v_2,v_3}$ being linearly independent implies ${v_1,v_2}$ is linearly independent as well.
answered Jan 21 at 22:38
JMoravitzJMoravitz
47.9k33886
answered Jan 21 at 22:38
JMoravitzJMoravitz
47.9k33886
answered Jan 21 at 22:38
JMoravitzJMoravitz
47.9k33886
answered Jan 21 at 22:38
JMoravitzJMoravitz
47.9k33886
47.9k33886
add a comment |
add a comment |
$begingroup$
IIRC,
if $v_{1,2,3}$
are linearly independent then
$sum a_iv_i = 0
implies
a_{1,2,3} = 0
$.
If
$v_1, v_2$
are not linearly independent,
there exist $a, b$
with at least one of
$a, b$ nonzero such that
$av_1+bv_2 =0$.
Therefore
$0
=av_1+bv_2 +0v_3$
so that
$v_{1, 2, 3}$
are not linearly independent.
Contradiction with assumption.
answered Jan 21 at 22:37
marty cohenmarty cohen
74k549128
$endgroup$
add a comment |
$begingroup$
IIRC,
if $v_{1,2,3}$
are linearly independent then
$sum a_iv_i = 0
implies
a_{1,2,3} = 0
$.
If
$v_1, v_2$
are not linearly independent,
there exist $a, b$
with at least one of
$a, b$ nonzero such that
$av_1+bv_2 =0$.
Therefore
$0
=av_1+bv_2 +0v_3$
so that
$v_{1, 2, 3}$
are not linearly independent.
Contradiction with assumption.
answered Jan 21 at 22:37
marty cohenmarty cohen
74k549128
$endgroup$
add a comment |
$begingroup$
IIRC,
if $v_{1,2,3}$
are linearly independent then
$sum a_iv_i = 0
implies
a_{1,2,3} = 0
$.
If
$v_1, v_2$
are not linearly independent,
there exist $a, b$
with at least one of
$a, b$ nonzero such that
$av_1+bv_2 =0$.
Therefore
$0
=av_1+bv_2 +0v_3$
so that
$v_{1, 2, 3}$
are not linearly independent.
Contradiction with assumption.
answered Jan 21 at 22:37
marty cohenmarty cohen
74k549128
$endgroup$
IIRC,
if $v_{1,2,3}$
are linearly independent then
$sum a_iv_i = 0
implies
a_{1,2,3} = 0
$.
If
$v_1, v_2$
are not linearly independent,
there exist $a, b$
with at least one of
$a, b$ nonzero such that
$av_1+bv_2 =0$.
Therefore
$0
=av_1+bv_2 +0v_3$
so that
$v_{1, 2, 3}$
are not linearly independent.
Contradiction with assumption.
answered Jan 21 at 22:37
marty cohenmarty cohen
74k549128
answered Jan 21 at 22:37
marty cohenmarty cohen
74k549128
answered Jan 21 at 22:37
marty cohenmarty cohen
74k549128
answered Jan 21 at 22:37
marty cohenmarty cohen
74k549128
74k549128
add a comment |
add a comment |
$begingroup$
Here's a short direct proof, without unnecessary negation:
(equivalent to the other answers)
Suppose
$a_1 v_1 + a_2 v_2 =0$.
We need to show $a_1=a_2=0$.
Since $0v_3=0$, we also have
$a_1 v_1 + a_2 v_2 + 0 v_3 =0$.
Since $v_1,v_2,v_3$ are linearly independent, $a_1=a_2=0$.
answered Jan 29 at 20:14
ffffforallffffforall
36028
$endgroup$
add a comment |
$begingroup$
Here's a short direct proof, without unnecessary negation:
(equivalent to the other answers)
Suppose
$a_1 v_1 + a_2 v_2 =0$.
We need to show $a_1=a_2=0$.
Since $0v_3=0$, we also have
$a_1 v_1 + a_2 v_2 + 0 v_3 =0$.
Since $v_1,v_2,v_3$ are linearly independent, $a_1=a_2=0$.
answered Jan 29 at 20:14
ffffforallffffforall
36028
$endgroup$
add a comment |
$begingroup$
Here's a short direct proof, without unnecessary negation:
(equivalent to the other answers)
Suppose
$a_1 v_1 + a_2 v_2 =0$.
We need to show $a_1=a_2=0$.
Since $0v_3=0$, we also have
$a_1 v_1 + a_2 v_2 + 0 v_3 =0$.
Since $v_1,v_2,v_3$ are linearly independent, $a_1=a_2=0$.
answered Jan 29 at 20:14
ffffforallffffforall
36028
$endgroup$
Here's a short direct proof, without unnecessary negation:
(equivalent to the other answers)
Suppose
$a_1 v_1 + a_2 v_2 =0$.
We need to show $a_1=a_2=0$.
Since $0v_3=0$, we also have
$a_1 v_1 + a_2 v_2 + 0 v_3 =0$.
Since $v_1,v_2,v_3$ are linearly independent, $a_1=a_2=0$.
answered Jan 29 at 20:14
ffffforallffffforall
36028
answered Jan 29 at 20:14
ffffforallffffforall
36028
answered Jan 29 at 20:14
ffffforallffffforall
36028
answered Jan 29 at 20:14
ffffforallffffforall
36028
36028
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%2f3082503%2fshow-that-if-vecv-1-vecv-2-vecv-3-is-a-linearly-independent-set%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
3
$begingroup$
Can you state the definitions of "${v_1, v_2, v_3}$ is a linearly independent set" and "${v_1, v_2}$ is a linearly independent set"? If you write out the definitions clearly, you should be able to finish the question in one step.
$endgroup$
– angryavian
Jan 21 at 22:28