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sswann222
 4 years ago
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.
sswann222
 4 years ago
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.

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JamesJ
 4 years ago
Best ResponseYou've already chosen the best response.1Three such vectorscall them p, q, rare 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 nonzero determinant. In other words, the determinant of the matrix who's entries are the components of p,q,r is nonzero. 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 nonzero determinant.

sswann222
 4 years ago
Best ResponseYou've already chosen the best response.0using that matrix i have to do row reduction methods?

JamesJ
 4 years ago
Best ResponseYou've already chosen the best response.1In which case, show that this matrix can be row reduced to the identity matrix. That's an equivalent statement.

sswann222
 4 years ago
Best ResponseYou've already chosen the best response.0I can get it to row reduced to the idenity which means that all three are linearly independent?....correct
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