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- Thread starter torquerotates
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morphism

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And of course any two vectors that span a plane are going to be linearly independent, just like any three vectors that span a 3-dimensional space are going to be linearly independent, or more generally, like any n vectors that span an n-dimensional space are going to be linearly independent (but any collection of more than n vectors isn't going to be).

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morphism

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Hope this clears things up for you.

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Thanks.

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thanks morphism for a really clear explanation

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Another way of looking at is, in an 'n' dimensional space maxminum number of linearly independent vectors is n, and any n linearly independent vectors (maximal linearly independent set) would be a basis for the space (of course spanning it).

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So how can I find out if a vector "v" in R3 is in the span of three other vectors, making up th columns of matrix "u" in R3?

Proposed solution:

1. determine if the vectors in u are dependent

yes: move on to step 2

no: u spans all R3 therefore any other vector in R3 is within the span

2. if the vectors in u are dependent, they span a plane in R3. Determine if v is linearly dependent WRT any two of the three vectors in u.

yes: the vector is within the span

no: the vector is not in the span

[I wish I could remove that reply from earlier. It was late, my mind was,.... well I'm not sure where it was. I was thinking, just stupidly.

Wait I can, sweet

I never made that stupid comment... really...]

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I've found my answer, I think. For anyone else who might need it....

From another https://www.physicsforums.com/showthread.php?t=193169&highlight=span":

Q"if any vector in P_3 can be written as a linear combo of the vectors in S, then can i conclude that the set S spans P_3?"

A:"Exactly!"

From another https://www.physicsforums.com/showthread.php?t=193169&highlight=span":

Q"if any vector in P_3 can be written as a linear combo of the vectors in S, then can i conclude that the set S spans P_3?"

A:"Exactly!"

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- #10

HallsofIvy

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That is, in fact, pretty much the definition of "span" of a set of vectors!

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