Geometric interpretation of a linear system Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 23, 2019 at 00:00UTC (8:00pm US/Eastern)Geometric Interpretation of Solutions to Linear SystemsSolution to this linear equation systemGeometric description of linear equationUnderstanding geometry of linear systemSystem of linear equations - ResolutionSolving Homogeneous System using Gauss–Jordan eliminationShow that the system of equations $Ax=b$ is not consistent for all $b$ in $mathbbR^3$Solving a simple system.Understanding solution of system of linear equationsProve the geometric interpretation?
What is the appropriate index architecture when forced to implement IsDeleted (soft deletes)?
Crossing US/Canada Border for less than 24 hours
Who can remove European Commissioners?
Aligning an equation at multiple points, with both left and right alignment, as well as equals sign alignment
What do you call the main part of a joke?
How to compare two different files line by line in unix?
How often does castling occur in grandmaster games?
How to dry out epoxy resin faster than usual?
What initially awakened the Balrog?
What to do with repeated rejections for phd position
Interpretation of R output from Cohen's Kappa
Strange behavior of Object.defineProperty() in JavaScript
Most bit efficient text communication method?
Can anything be seen from the center of the Boötes void? How dark would it be?
Why weren't discrete x86 CPUs ever used in game hardware?
How does light 'choose' between wave and particle behaviour?
Amount of permutations on an NxNxN Rubik's Cube
An adverb for when you're not exaggerating
Belief In God or Knowledge Of God. Which is better?
Put R under double integral
Quadrilaterals with equal sides
Why is it faster to reheat something than it is to cook it?
How do living politicians protect their readily obtainable signatures from misuse?
Did Mueller's report provide an evidentiary basis for the claim of Russian govt election interference via social media?
Geometric interpretation of a linear system
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 23, 2019 at 00:00UTC (8:00pm US/Eastern)Geometric Interpretation of Solutions to Linear SystemsSolution to this linear equation systemGeometric description of linear equationUnderstanding geometry of linear systemSystem of linear equations - ResolutionSolving Homogeneous System using Gauss–Jordan eliminationShow that the system of equations $Ax=b$ is not consistent for all $b$ in $mathbbR^3$Solving a simple system.Understanding solution of system of linear equationsProve the geometric interpretation?
$begingroup$
Solve the following system of linear equations in terms of parameter $ainmathbb R$ and explain geometric interpretation of this system:
$ax+y+z=1,2x+2ay+2z=3, x+y+az=1$.
By Cronecker Capelli's theorem, we get:
$$ beginbmatrix
a & 1 & 1 & 1\
2 & 2a & 2 & 3\
1 & 1 & a & 1\
endbmatrix$$
Row echelon form of this matrix is $$beginbmatrix
1 & 1 & a & 1\
0 & 2(a-1) & 2(1-a) & 1\
0 & 0 & (1-a)(a+2) & (3-2a)/2\
endbmatrix$$
System is inconsistent for $a=1 lor a=-2 $. For every other value of $a$, system has unique solution.
For every $a$ instead of $a=1 land a=-2$ there are three planes that intersect at a point.
Question: Is this geometric interpretation correct?
linear-algebra matrices systems-of-equations plane-geometry
$endgroup$
add a comment |
$begingroup$
Solve the following system of linear equations in terms of parameter $ainmathbb R$ and explain geometric interpretation of this system:
$ax+y+z=1,2x+2ay+2z=3, x+y+az=1$.
By Cronecker Capelli's theorem, we get:
$$ beginbmatrix
a & 1 & 1 & 1\
2 & 2a & 2 & 3\
1 & 1 & a & 1\
endbmatrix$$
Row echelon form of this matrix is $$beginbmatrix
1 & 1 & a & 1\
0 & 2(a-1) & 2(1-a) & 1\
0 & 0 & (1-a)(a+2) & (3-2a)/2\
endbmatrix$$
System is inconsistent for $a=1 lor a=-2 $. For every other value of $a$, system has unique solution.
For every $a$ instead of $a=1 land a=-2$ there are three planes that intersect at a point.
Question: Is this geometric interpretation correct?
linear-algebra matrices systems-of-equations plane-geometry
$endgroup$
$begingroup$
You should also study what relation exists between the planes when $a=1$ and when $a=-2$. Are they parallel? Are two of them parallel and does one of them intersect with these two?...
$endgroup$
– Bernard Masse
Jul 16 '16 at 17:28
add a comment |
$begingroup$
Solve the following system of linear equations in terms of parameter $ainmathbb R$ and explain geometric interpretation of this system:
$ax+y+z=1,2x+2ay+2z=3, x+y+az=1$.
By Cronecker Capelli's theorem, we get:
$$ beginbmatrix
a & 1 & 1 & 1\
2 & 2a & 2 & 3\
1 & 1 & a & 1\
endbmatrix$$
Row echelon form of this matrix is $$beginbmatrix
1 & 1 & a & 1\
0 & 2(a-1) & 2(1-a) & 1\
0 & 0 & (1-a)(a+2) & (3-2a)/2\
endbmatrix$$
System is inconsistent for $a=1 lor a=-2 $. For every other value of $a$, system has unique solution.
For every $a$ instead of $a=1 land a=-2$ there are three planes that intersect at a point.
Question: Is this geometric interpretation correct?
linear-algebra matrices systems-of-equations plane-geometry
$endgroup$
Solve the following system of linear equations in terms of parameter $ainmathbb R$ and explain geometric interpretation of this system:
$ax+y+z=1,2x+2ay+2z=3, x+y+az=1$.
By Cronecker Capelli's theorem, we get:
$$ beginbmatrix
a & 1 & 1 & 1\
2 & 2a & 2 & 3\
1 & 1 & a & 1\
endbmatrix$$
Row echelon form of this matrix is $$beginbmatrix
1 & 1 & a & 1\
0 & 2(a-1) & 2(1-a) & 1\
0 & 0 & (1-a)(a+2) & (3-2a)/2\
endbmatrix$$
System is inconsistent for $a=1 lor a=-2 $. For every other value of $a$, system has unique solution.
For every $a$ instead of $a=1 land a=-2$ there are three planes that intersect at a point.
Question: Is this geometric interpretation correct?
linear-algebra matrices systems-of-equations plane-geometry
linear-algebra matrices systems-of-equations plane-geometry
asked Jul 16 '16 at 17:20
user300046user300046
386
386
$begingroup$
You should also study what relation exists between the planes when $a=1$ and when $a=-2$. Are they parallel? Are two of them parallel and does one of them intersect with these two?...
$endgroup$
– Bernard Masse
Jul 16 '16 at 17:28
add a comment |
$begingroup$
You should also study what relation exists between the planes when $a=1$ and when $a=-2$. Are they parallel? Are two of them parallel and does one of them intersect with these two?...
$endgroup$
– Bernard Masse
Jul 16 '16 at 17:28
$begingroup$
You should also study what relation exists between the planes when $a=1$ and when $a=-2$. Are they parallel? Are two of them parallel and does one of them intersect with these two?...
$endgroup$
– Bernard Masse
Jul 16 '16 at 17:28
$begingroup$
You should also study what relation exists between the planes when $a=1$ and when $a=-2$. Are they parallel? Are two of them parallel and does one of them intersect with these two?...
$endgroup$
– Bernard Masse
Jul 16 '16 at 17:28
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
When $aneq1$ and $aneq-2$ the solution is $$x=z=frac2a-32(a-1)(a+2),qquad y=frac3a-12(a-1)(a+2)$$
When $a=1$ the first and third equations are the same and inconsistent with the second one, i.e. the plane described by the first and third equations is parallel to the plane defined by the second equation. The augmented matrix has rank 2 while the coefficient matrix has rank 1.
When $a=-2$ the augmented matrix has rank $3$ and the coefficient matrix has rank 2, so again the three equations are inconsistent. You might find this page useful to work out the geometric interpretation: http://www.vitutor.com/geometry/space/three_planes.html
$endgroup$
$begingroup$
To summarize, in case where $aneq 1 land aneq -2$ we have two coincident planes and one that intersects them in a line. In case where $a=1 land a=-2$ we have two coincident planes and the other is parallel to them. Is this correct?
$endgroup$
– user300046
Jul 16 '16 at 18:41
$begingroup$
No, when $aneq 1$ and $aneq 2$, both the augmented matrix and coefficient matrix have rank 3 and so the planes intersect at a point (the one given in my answer). When $a=1$ the two of the planes coincide and are parallel the the third plane. When $a=-2$ each plane cuts the other two in a line. Look at the link in my answer.
$endgroup$
– smcc
Jul 16 '16 at 21:23
add a comment |
Your Answer
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%2f1861440%2fgeometric-interpretation-of-a-linear-system%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
$begingroup$
When $aneq1$ and $aneq-2$ the solution is $$x=z=frac2a-32(a-1)(a+2),qquad y=frac3a-12(a-1)(a+2)$$
When $a=1$ the first and third equations are the same and inconsistent with the second one, i.e. the plane described by the first and third equations is parallel to the plane defined by the second equation. The augmented matrix has rank 2 while the coefficient matrix has rank 1.
When $a=-2$ the augmented matrix has rank $3$ and the coefficient matrix has rank 2, so again the three equations are inconsistent. You might find this page useful to work out the geometric interpretation: http://www.vitutor.com/geometry/space/three_planes.html
$endgroup$
$begingroup$
To summarize, in case where $aneq 1 land aneq -2$ we have two coincident planes and one that intersects them in a line. In case where $a=1 land a=-2$ we have two coincident planes and the other is parallel to them. Is this correct?
$endgroup$
– user300046
Jul 16 '16 at 18:41
$begingroup$
No, when $aneq 1$ and $aneq 2$, both the augmented matrix and coefficient matrix have rank 3 and so the planes intersect at a point (the one given in my answer). When $a=1$ the two of the planes coincide and are parallel the the third plane. When $a=-2$ each plane cuts the other two in a line. Look at the link in my answer.
$endgroup$
– smcc
Jul 16 '16 at 21:23
add a comment |
$begingroup$
When $aneq1$ and $aneq-2$ the solution is $$x=z=frac2a-32(a-1)(a+2),qquad y=frac3a-12(a-1)(a+2)$$
When $a=1$ the first and third equations are the same and inconsistent with the second one, i.e. the plane described by the first and third equations is parallel to the plane defined by the second equation. The augmented matrix has rank 2 while the coefficient matrix has rank 1.
When $a=-2$ the augmented matrix has rank $3$ and the coefficient matrix has rank 2, so again the three equations are inconsistent. You might find this page useful to work out the geometric interpretation: http://www.vitutor.com/geometry/space/three_planes.html
$endgroup$
$begingroup$
To summarize, in case where $aneq 1 land aneq -2$ we have two coincident planes and one that intersects them in a line. In case where $a=1 land a=-2$ we have two coincident planes and the other is parallel to them. Is this correct?
$endgroup$
– user300046
Jul 16 '16 at 18:41
$begingroup$
No, when $aneq 1$ and $aneq 2$, both the augmented matrix and coefficient matrix have rank 3 and so the planes intersect at a point (the one given in my answer). When $a=1$ the two of the planes coincide and are parallel the the third plane. When $a=-2$ each plane cuts the other two in a line. Look at the link in my answer.
$endgroup$
– smcc
Jul 16 '16 at 21:23
add a comment |
$begingroup$
When $aneq1$ and $aneq-2$ the solution is $$x=z=frac2a-32(a-1)(a+2),qquad y=frac3a-12(a-1)(a+2)$$
When $a=1$ the first and third equations are the same and inconsistent with the second one, i.e. the plane described by the first and third equations is parallel to the plane defined by the second equation. The augmented matrix has rank 2 while the coefficient matrix has rank 1.
When $a=-2$ the augmented matrix has rank $3$ and the coefficient matrix has rank 2, so again the three equations are inconsistent. You might find this page useful to work out the geometric interpretation: http://www.vitutor.com/geometry/space/three_planes.html
$endgroup$
When $aneq1$ and $aneq-2$ the solution is $$x=z=frac2a-32(a-1)(a+2),qquad y=frac3a-12(a-1)(a+2)$$
When $a=1$ the first and third equations are the same and inconsistent with the second one, i.e. the plane described by the first and third equations is parallel to the plane defined by the second equation. The augmented matrix has rank 2 while the coefficient matrix has rank 1.
When $a=-2$ the augmented matrix has rank $3$ and the coefficient matrix has rank 2, so again the three equations are inconsistent. You might find this page useful to work out the geometric interpretation: http://www.vitutor.com/geometry/space/three_planes.html
edited Jul 16 '16 at 17:44
answered Jul 16 '16 at 17:28
smccsmcc
4,317517
4,317517
$begingroup$
To summarize, in case where $aneq 1 land aneq -2$ we have two coincident planes and one that intersects them in a line. In case where $a=1 land a=-2$ we have two coincident planes and the other is parallel to them. Is this correct?
$endgroup$
– user300046
Jul 16 '16 at 18:41
$begingroup$
No, when $aneq 1$ and $aneq 2$, both the augmented matrix and coefficient matrix have rank 3 and so the planes intersect at a point (the one given in my answer). When $a=1$ the two of the planes coincide and are parallel the the third plane. When $a=-2$ each plane cuts the other two in a line. Look at the link in my answer.
$endgroup$
– smcc
Jul 16 '16 at 21:23
add a comment |
$begingroup$
To summarize, in case where $aneq 1 land aneq -2$ we have two coincident planes and one that intersects them in a line. In case where $a=1 land a=-2$ we have two coincident planes and the other is parallel to them. Is this correct?
$endgroup$
– user300046
Jul 16 '16 at 18:41
$begingroup$
No, when $aneq 1$ and $aneq 2$, both the augmented matrix and coefficient matrix have rank 3 and so the planes intersect at a point (the one given in my answer). When $a=1$ the two of the planes coincide and are parallel the the third plane. When $a=-2$ each plane cuts the other two in a line. Look at the link in my answer.
$endgroup$
– smcc
Jul 16 '16 at 21:23
$begingroup$
To summarize, in case where $aneq 1 land aneq -2$ we have two coincident planes and one that intersects them in a line. In case where $a=1 land a=-2$ we have two coincident planes and the other is parallel to them. Is this correct?
$endgroup$
– user300046
Jul 16 '16 at 18:41
$begingroup$
To summarize, in case where $aneq 1 land aneq -2$ we have two coincident planes and one that intersects them in a line. In case where $a=1 land a=-2$ we have two coincident planes and the other is parallel to them. Is this correct?
$endgroup$
– user300046
Jul 16 '16 at 18:41
$begingroup$
No, when $aneq 1$ and $aneq 2$, both the augmented matrix and coefficient matrix have rank 3 and so the planes intersect at a point (the one given in my answer). When $a=1$ the two of the planes coincide and are parallel the the third plane. When $a=-2$ each plane cuts the other two in a line. Look at the link in my answer.
$endgroup$
– smcc
Jul 16 '16 at 21:23
$begingroup$
No, when $aneq 1$ and $aneq 2$, both the augmented matrix and coefficient matrix have rank 3 and so the planes intersect at a point (the one given in my answer). When $a=1$ the two of the planes coincide and are parallel the the third plane. When $a=-2$ each plane cuts the other two in a line. Look at the link in my answer.
$endgroup$
– smcc
Jul 16 '16 at 21:23
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%2f1861440%2fgeometric-interpretation-of-a-linear-system%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
$begingroup$
You should also study what relation exists between the planes when $a=1$ and when $a=-2$. Are they parallel? Are two of them parallel and does one of them intersect with these two?...
$endgroup$
– Bernard Masse
Jul 16 '16 at 17:28