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Geometric interpretation of a linear system

0

$begingroup$

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

share|cite|improve this question

asked Jul 16 '16 at 17:20

user300046user300046

386

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  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 | 

0

$begingroup$

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

share|cite|improve this question

asked Jul 16 '16 at 17:20

user300046user300046

386

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  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

share|cite|improve this question

asked Jul 16 '16 at 17:20

user300046user300046

386

$endgroup$

share|cite|improve this question

asked Jul 16 '16 at 17:20

user300046user300046

386

share|cite|improve this question

asked Jul 16 '16 at 17:20

user300046user300046

386

asked Jul 16 '16 at 17:20

user300046user300046

386

asked Jul 16 '16 at 17:20

user300046user300046

386

1 Answer
1

active

oldest

votes

0

$begingroup$

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

share|cite|improve this answer

edited Jul 16 '16 at 17:44

answered Jul 16 '16 at 17:28

smccsmcc

4,317517

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  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 | 

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0

$begingroup$

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

share|cite|improve this answer

edited Jul 16 '16 at 17:44

answered Jul 16 '16 at 17:28

smccsmcc

4,317517

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  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 | 

0

$begingroup$

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

share|cite|improve this answer

edited Jul 16 '16 at 17:44

answered Jul 16 '16 at 17:28

smccsmcc

4,317517

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  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

share|cite|improve this answer

edited Jul 16 '16 at 17:44

answered Jul 16 '16 at 17:28

smccsmcc

4,317517

$endgroup$

share|cite|improve this answer

edited Jul 16 '16 at 17:44

answered Jul 16 '16 at 17:28

smccsmcc

4,317517

answered Jul 16 '16 at 17:28

smccsmcc

4,317517

answered Jul 16 '16 at 17:28

smccsmcc

4,317517