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Finding correct projective change of coordinates $M$.

0

$begingroup$

Find a $3 times 3$ matrix $M$ such that, under the change of varibles
$$beginbmatrix
x \
y \
z
endbmatrix = M^-1beginbmatrix
x_1 \
y_1 \
z_1
endbmatrix,$$
we have $(1 : 0 : 1) mapsto (0: 1 : 0)$ and $l(x, y, z) = x - z$ becoming $l_1(x_1, y_1, z_1) = z_1$.

My strategy so far has been to replace $M^-1$ with $beginbmatrix
a & b & c \
d & e & f \
g & h & i
endbmatrix$, so that

$$beginbmatrix
1 \
0 \
1
endbmatrix = beginbmatrix
a & b & c \
d & e & f \
g & h & i
endbmatrixbeginbmatrix
0 \
1 \
0
endbmatrix,$$
which implies $b = 1, e = 0,$ and $h = 1$. However, I don't know how to use the rest of the information to find the remaining unknowns.

matrices algebraic-geometry projective-geometry projective-space change-of-variable

share|cite|improve this question

asked Mar 26 at 17:16

SmashSmash

1417

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Be careful. You’re dealing with homogeneous vectors and matrices, so you can’t insist on strict equality after multiplying by the transformation matrix.
  $endgroup$
  – amd
  Mar 26 at 22:54
  
  

  
  
  
  

add a comment | 

0

$begingroup$

Find a $3 times 3$ matrix $M$ such that, under the change of varibles
$$beginbmatrix
x \
y \
z
endbmatrix = M^-1beginbmatrix
x_1 \
y_1 \
z_1
endbmatrix,$$
we have $(1 : 0 : 1) mapsto (0: 1 : 0)$ and $l(x, y, z) = x - z$ becoming $l_1(x_1, y_1, z_1) = z_1$.

My strategy so far has been to replace $M^-1$ with $beginbmatrix
a & b & c \
d & e & f \
g & h & i
endbmatrix$, so that

$$beginbmatrix
1 \
0 \
1
endbmatrix = beginbmatrix
a & b & c \
d & e & f \
g & h & i
endbmatrixbeginbmatrix
0 \
1 \
0
endbmatrix,$$
which implies $b = 1, e = 0,$ and $h = 1$. However, I don't know how to use the rest of the information to find the remaining unknowns.

matrices algebraic-geometry projective-geometry projective-space change-of-variable

share|cite|improve this question

asked Mar 26 at 17:16

SmashSmash

1417

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Be careful. You’re dealing with homogeneous vectors and matrices, so you can’t insist on strict equality after multiplying by the transformation matrix.
  $endgroup$
  – amd
  Mar 26 at 22:54
  
  

  
  
  
  

add a comment | 

$begingroup$

Find a $3 times 3$ matrix $M$ such that, under the change of varibles
$$beginbmatrix
x \
y \
z
endbmatrix = M^-1beginbmatrix
x_1 \
y_1 \
z_1
endbmatrix,$$
we have $(1 : 0 : 1) mapsto (0: 1 : 0)$ and $l(x, y, z) = x - z$ becoming $l_1(x_1, y_1, z_1) = z_1$.

My strategy so far has been to replace $M^-1$ with $beginbmatrix
a & b & c \
d & e & f \
g & h & i
endbmatrix$, so that

$$beginbmatrix
1 \
0 \
1
endbmatrix = beginbmatrix
a & b & c \
d & e & f \
g & h & i
endbmatrixbeginbmatrix
0 \
1 \
0
endbmatrix,$$
which implies $b = 1, e = 0,$ and $h = 1$. However, I don't know how to use the rest of the information to find the remaining unknowns.

matrices algebraic-geometry projective-geometry projective-space change-of-variable

share|cite|improve this question

asked Mar 26 at 17:16

SmashSmash

1417

$endgroup$

share|cite|improve this question

asked Mar 26 at 17:16

SmashSmash

1417

share|cite|improve this question

asked Mar 26 at 17:16

SmashSmash

1417

asked Mar 26 at 17:16

SmashSmash

1417

asked Mar 26 at 17:16

SmashSmash

1417