sswann222
  • sswann222
how do i prove linear independence? (2,1,-2)^T, (3,2,-1)^T, (2,2,0)^T..given these three of vectors of a vector space.
Mathematics
schrodinger
  • schrodinger
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JamesJ
  • JamesJ
Three such vectors--call them p, q, r--are linearly independent if you cannot write one as a sum of the others. That is equivalent to the idea that the equation ap + bp + cr = 0 has only one solution, the trivial solution: a=b=c=0. And that is equivalent to the statement the corresponding system of linear equations in a,b,c only has one solution: a=b=c=0. And that is equivalent to the statement that if you put the coefficients of those equations in a matrix, that matrix has non-zero determinant. In other words, the determinant of the matrix who's entries are the components of p,q,r is non-zero. Hence to show that p,q,r are linearly independent, is sufficient to show that the matrix 2 3 2 1 2 2 -2 -1 0 has non-zero determinant.
sswann222
  • sswann222
using that matrix i have to do row reduction methods?
JamesJ
  • JamesJ
In which case, show that this matrix can be row reduced to the identity matrix. That's an equivalent statement.

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sswann222
  • sswann222
I can get it to row reduced to the idenity which means that all three are linearly independent?....correct
JamesJ
  • JamesJ
Yes.
sswann222
  • sswann222
thank you

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