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i am just stuck now on how to get this proof started... any thoughts on how to start?

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In summary, when proving that if L and P are three dimensional subspaces of R5, they must have a nonzero constant in common, it is important to consider whether the basis vectors of these subspaces are linearly independent. If not, then they may not span the space and therefore will not have a nonzero constant in common.

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i am just stuck now on how to get this proof started... any thoughts on how to start?

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thanks a lot!

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they are not linearly independent? and thus not a basis for such space

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If your 6 basis vectors are not linearly independent, then they share...

What does it mean if the vectors are not linearly independent?

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then they don't share common subspaces...

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matt grime said:

gavin1989 said:

thanks a lot!

matt grime said:

Notice that you did not use the fact that the two subspaces have no non-zero vector (you said "constant" but I presume you meant vector) in common. That's the important thing.

Having a common nonzero constant means that there is a number that can be multiplied by all the elements in both L and P, resulting in the same value. This number is not equal to zero, as that would mean the two sets have no common elements.

R^{5} refers to the 5-dimensional Euclidean space, which is the space in which the elements of L and P exist. This proof shows that despite being in a high-dimensional space, L and P still have a common nonzero constant, demonstrating a fundamental property of vector spaces.

This proof has implications in linear algebra and other areas of mathematics. It shows that even in high-dimensional spaces, certain properties and relationships still hold true. It also allows for the use of techniques and methods from linear algebra in higher dimensions.

Many real-world systems can be represented and analyzed using vectors in high-dimensional spaces. This proof provides a fundamental understanding of the properties and relationships of these vectors, which can be applied to various fields such as physics, engineering, and data analysis.

Yes, this proof applies to all sets in R^{5} as long as the sets are linearly dependent. However, if the sets are linearly independent, then they cannot have a common nonzero constant. This proof only holds true for linearly dependent sets in R^{5}.

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