Finding correct projective change of coordinates $M$. Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Why does it take 5 points to construct a projective frame in $mathbbR^4$?Finding eigenvalues of a matrix with two unknownsDoes this operation exist? What's its name?Equation for 3D vector rotationFinding matrix tranformationHow to transform multiple 3d points by different distances with a matix calculation?Affine coordinates and projective spacesChange of coordinates between basesDisjoint Lines in Projective SpaceFind a Givens rotation matrix such that $y=Gx$

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Finding correct projective change of coordinates $M$.



Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Why does it take 5 points to construct a projective frame in $mathbbR^4$?Finding eigenvalues of a matrix with two unknownsDoes this operation exist? What's its name?Equation for 3D vector rotationFinding matrix tranformationHow to transform multiple 3d points by different distances with a matix calculation?Affine coordinates and projective spacesChange of coordinates between basesDisjoint Lines in Projective SpaceFind a Givens rotation matrix such that $y=Gx$










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.










share|cite|improve this question









$endgroup$











  • $begingroup$
    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
















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.










share|cite|improve this question









$endgroup$











  • $begingroup$
    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














0












0








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.










share|cite|improve this question









$endgroup$




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













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Mar 26 at 17:16









SmashSmash

1417




1417











  • $begingroup$
    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

















  • $begingroup$
    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
















$begingroup$
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





$begingroup$
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











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