Question about projective line and intersection
Clash Royale CLAN TAG#URR8PPP
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The book I'm reading through stated the following:
Let $mathbb P^1$ be the complex projective line, pick a random line that doesn't pass through the origin in $mathbb C^2$. Then any point $pinmathbb P^1$ (i.e. a line through origin) will intersect the affine line at exactly one unique point, except the parallel line, which we denote as the $infty$.
I guess my question is about how to visualize complex vector spaces. I see how the above would work for $mathbb R$. Two lines will intersect at exactly a point. But for the case of $mathbb C$, while I can tell why two complex lines intersect at a point by linear algebra, my doubt is both the one-dimensional subspace and the affine line basically "look like planes", and so their intersection should "look like a line". My question is how does this correspond to a point? My speculation is that thinking $mathbb C^2$ as something like $mathbb R^4$ is the wrong way to visualize it.
projective-space visualization
asked Aug 24 at 2:33
lEm
3,0581618
add a comment |Â
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0
down vote
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The book I'm reading through stated the following:
Let $mathbb P^1$ be the complex projective line, pick a random line that doesn't pass through the origin in $mathbb C^2$. Then any point $pinmathbb P^1$ (i.e. a line through origin) will intersect the affine line at exactly one unique point, except the parallel line, which we denote as the $infty$.
I guess my question is about how to visualize complex vector spaces. I see how the above would work for $mathbb R$. Two lines will intersect at exactly a point. But for the case of $mathbb C$, while I can tell why two complex lines intersect at a point by linear algebra, my doubt is both the one-dimensional subspace and the affine line basically "look like planes", and so their intersection should "look like a line". My question is how does this correspond to a point? My speculation is that thinking $mathbb C^2$ as something like $mathbb R^4$ is the wrong way to visualize it.
projective-space visualization
asked Aug 24 at 2:33
lEm
3,0581618
In $mathbb R^4$ it is possible for two planes to intersect in a single point (e.g. consider the planes Danner by the coordinate axes).
– asdq
Aug 24 at 5:13
@asdq ah I see. That was so obvious that I couldn't believe I missed that.
– lEm
Aug 24 at 11:28
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
The book I'm reading through stated the following:
Let $mathbb P^1$ be the complex projective line, pick a random line that doesn't pass through the origin in $mathbb C^2$. Then any point $pinmathbb P^1$ (i.e. a line through origin) will intersect the affine line at exactly one unique point, except the parallel line, which we denote as the $infty$.
I guess my question is about how to visualize complex vector spaces. I see how the above would work for $mathbb R$. Two lines will intersect at exactly a point. But for the case of $mathbb C$, while I can tell why two complex lines intersect at a point by linear algebra, my doubt is both the one-dimensional subspace and the affine line basically "look like planes", and so their intersection should "look like a line". My question is how does this correspond to a point? My speculation is that thinking $mathbb C^2$ as something like $mathbb R^4$ is the wrong way to visualize it.
projective-space visualization
asked Aug 24 at 2:33
lEm
3,0581618
The book I'm reading through stated the following:
Let $mathbb P^1$ be the complex projective line, pick a random line that doesn't pass through the origin in $mathbb C^2$. Then any point $pinmathbb P^1$ (i.e. a line through origin) will intersect the affine line at exactly one unique point, except the parallel line, which we denote as the $infty$.
I guess my question is about how to visualize complex vector spaces. I see how the above would work for $mathbb R$. Two lines will intersect at exactly a point. But for the case of $mathbb C$, while I can tell why two complex lines intersect at a point by linear algebra, my doubt is both the one-dimensional subspace and the affine line basically "look like planes", and so their intersection should "look like a line". My question is how does this correspond to a point? My speculation is that thinking $mathbb C^2$ as something like $mathbb R^4$ is the wrong way to visualize it.
projective-space visualization
asked Aug 24 at 2:33
lEm
3,0581618
asked Aug 24 at 2:33
lEm
3,0581618
asked Aug 24 at 2:33
lEm
3,0581618
asked Aug 24 at 2:33
lEm
3,0581618
3,0581618
In $mathbb R^4$ it is possible for two planes to intersect in a single point (e.g. consider the planes Danner by the coordinate axes).
– asdq
Aug 24 at 5:13
@asdq ah I see. That was so obvious that I couldn't believe I missed that.
– lEm
Aug 24 at 11:28
add a comment |Â
In $mathbb R^4$ it is possible for two planes to intersect in a single point (e.g. consider the planes Danner by the coordinate axes).
– asdq
Aug 24 at 5:13
@asdq ah I see. That was so obvious that I couldn't believe I missed that.
– lEm
Aug 24 at 11:28
In $mathbb R^4$ it is possible for two planes to intersect in a single point (e.g. consider the planes Danner by the coordinate axes).
– asdq
Aug 24 at 5:13
In $mathbb R^4$ it is possible for two planes to intersect in a single point (e.g. consider the planes Danner by the coordinate axes).
– asdq
Aug 24 at 5:13
@asdq ah I see. That was so obvious that I couldn't believe I missed that.
– lEm
Aug 24 at 11:28
@asdq ah I see. That was so obvious that I couldn't believe I missed that.
– lEm
Aug 24 at 11:28
add a comment |Â
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In $mathbb R^4$ it is possible for two planes to intersect in a single point (e.g. consider the planes Danner by the coordinate axes).
– asdq
Aug 24 at 5:13
@asdq ah I see. That was so obvious that I couldn't believe I missed that.
– lEm
Aug 24 at 11:28