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Do concept of linear independence exist in geometry?
The Next CEO of Stack OverflowUsing the Determinant to verify Linear Independence, Span and BasisWhat are matroids and in what cases are they useful?Proving linear independence of infinite set (monomials)Differences between finding linear independence and finding a spanning set.Linear independence, Spanning, Injectivity, and SurjectivityWhen does non-spanning imply linear independence in a free $R$-moduleGood book on non-Eucledian geometry with linear algebra approachFor bases, why bother with span?The Basis of a set of Linear TransformationsLinear algebra concept clarification
$begingroup$
Just like in linear algebra we encounter concept of linear independence spanning set. Does there exist any such concept in geometry? (For example can a circle and straight line span a $mathbb R^2$ space?)
linear-algebra geometry algebraic-geometry
asked Mar 27 at 18:14
Bijayan RayBijayan Ray
1431213
$endgroup$
add a comment |
$begingroup$
Just like in linear algebra we encounter concept of linear independence spanning set. Does there exist any such concept in geometry? (For example can a circle and straight line span a $mathbb R^2$ space?)
linear-algebra geometry algebraic-geometry
asked Mar 27 at 18:14
Bijayan RayBijayan Ray
1431213
$endgroup$
2
$begingroup$
If you want to think about the span of two vectors as $mathbbRtimesmathbbRcongmathbbR^2$, then perhaps you want the “span” of a circle and a line to be a cylinder $S^1timesmathbbR$. However, geometry is a bit more complicated because a Mobius strip is also a circle with lines attached, so this starts to feel like a discussion of fiber bundles...
$endgroup$
– Santana Afton
Mar 27 at 18:33
$begingroup$
@SantanaAfton I have a very little knowledge in geometry, the question just came to my mind, I would like to know under which branch or chapters of mathematics do these kind of questions occurs? Actually I have not understood your comment completely well, like what is $S^1$ ?
$endgroup$
– Bijayan Ray
2 days ago
add a comment |
$begingroup$
Just like in linear algebra we encounter concept of linear independence spanning set. Does there exist any such concept in geometry? (For example can a circle and straight line span a $mathbb R^2$ space?)
linear-algebra geometry algebraic-geometry
asked Mar 27 at 18:14
Bijayan RayBijayan Ray
1431213
$endgroup$
Just like in linear algebra we encounter concept of linear independence spanning set. Does there exist any such concept in geometry? (For example can a circle and straight line span a $mathbb R^2$ space?)
linear-algebra geometry algebraic-geometry
linear-algebra geometry algebraic-geometry
asked Mar 27 at 18:14
Bijayan RayBijayan Ray
1431213
asked Mar 27 at 18:14
Bijayan RayBijayan Ray
1431213
asked Mar 27 at 18:14
Bijayan RayBijayan Ray
1431213
asked Mar 27 at 18:14
Bijayan RayBijayan Ray
1431213
asked Mar 27 at 18:14
Bijayan RayBijayan Ray
1431213
1431213
2
$begingroup$
If you want to think about the span of two vectors as $mathbbRtimesmathbbRcongmathbbR^2$, then perhaps you want the “span” of a circle and a line to be a cylinder $S^1timesmathbbR$. However, geometry is a bit more complicated because a Mobius strip is also a circle with lines attached, so this starts to feel like a discussion of fiber bundles...
$endgroup$
– Santana Afton
Mar 27 at 18:33
$begingroup$
@SantanaAfton I have a very little knowledge in geometry, the question just came to my mind, I would like to know under which branch or chapters of mathematics do these kind of questions occurs? Actually I have not understood your comment completely well, like what is $S^1$ ?
$endgroup$
– Bijayan Ray
2 days ago
add a comment |
2
$begingroup$
If you want to think about the span of two vectors as $mathbbRtimesmathbbRcongmathbbR^2$, then perhaps you want the “span” of a circle and a line to be a cylinder $S^1timesmathbbR$. However, geometry is a bit more complicated because a Mobius strip is also a circle with lines attached, so this starts to feel like a discussion of fiber bundles...
$endgroup$
– Santana Afton
Mar 27 at 18:33
$begingroup$
@SantanaAfton I have a very little knowledge in geometry, the question just came to my mind, I would like to know under which branch or chapters of mathematics do these kind of questions occurs? Actually I have not understood your comment completely well, like what is $S^1$ ?
$endgroup$
– Bijayan Ray
2 days ago
2
2
$begingroup$
If you want to think about the span of two vectors as $mathbbRtimesmathbbRcongmathbbR^2$, then perhaps you want the “span” of a circle and a line to be a cylinder $S^1timesmathbbR$. However, geometry is a bit more complicated because a Mobius strip is also a circle with lines attached, so this starts to feel like a discussion of fiber bundles...
$endgroup$
– Santana Afton
Mar 27 at 18:33
$begingroup$
If you want to think about the span of two vectors as $mathbbRtimesmathbbRcongmathbbR^2$, then perhaps you want the “span” of a circle and a line to be a cylinder $S^1timesmathbbR$. However, geometry is a bit more complicated because a Mobius strip is also a circle with lines attached, so this starts to feel like a discussion of fiber bundles...
$endgroup$
– Santana Afton
Mar 27 at 18:33
$begingroup$
@SantanaAfton I have a very little knowledge in geometry, the question just came to my mind, I would like to know under which branch or chapters of mathematics do these kind of questions occurs? Actually I have not understood your comment completely well, like what is $S^1$ ?
$endgroup$
– Bijayan Ray
2 days ago
$begingroup$
@SantanaAfton I have a very little knowledge in geometry, the question just came to my mind, I would like to know under which branch or chapters of mathematics do these kind of questions occurs? Actually I have not understood your comment completely well, like what is $S^1$ ?
$endgroup$
– Bijayan Ray
2 days ago
add a comment |
0
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$begingroup$
If you want to think about the span of two vectors as $mathbbRtimesmathbbRcongmathbbR^2$, then perhaps you want the “span” of a circle and a line to be a cylinder $S^1timesmathbbR$. However, geometry is a bit more complicated because a Mobius strip is also a circle with lines attached, so this starts to feel like a discussion of fiber bundles...
$endgroup$
– Santana Afton
Mar 27 at 18:33
$begingroup$
@SantanaAfton I have a very little knowledge in geometry, the question just came to my mind, I would like to know under which branch or chapters of mathematics do these kind of questions occurs? Actually I have not understood your comment completely well, like what is $S^1$ ?
$endgroup$
– Bijayan Ray
2 days ago