Proving Linear Independence: Vectors in R^5 and Their Span

Thread starter NeonVomitt
Start date Mar 3, 2009
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Vectors

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To determine if at least one of the vectors v1, v2, or v3 is not in the span of vectors a1, a2, and a3, one can analyze the transformation matrix T formed by the coefficients of a1, a2, and a3. Since v1, v2, and v3 are linearly independent in R^5, the rank of the transformation matrix T must be less than 3 for the vectors a1, a2, and a3 to span a space that does not include all of v1, v2, and v3. Reducing T to row echelon form will reveal its rank, helping to confirm whether the span of a1, a2, and a3 can encompass all of v1, v2, and v3. If the rank of T is less than 3, it indicates that at least one of the original vectors is excluded from the span. Therefore, performing the row reduction is essential for proving the claim.

Mar 3, 2009

NeonVomitt

Suppose that v1,v2,v3 are linearly independent vectors in R^5 and consider the vectors a1,a2,a3 defined by a1=v1+v2-2v3, a2=3v1+v2+4va, and a3=v1+2v2-7v3. Show that at least one of the vectors v1,v2,v3 is not in the span of the vectors a1,a2,a3.

I am kind of confused. Should I somehow reduce row echelon it? But how would I even set that up given this type of format?

Thank you!

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Mar 3, 2009

lurflurf

Homework Helper

a=T.v
where
T={{1,1,-2},{3,1,4},{1,2,-7}}

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