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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.

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  1. JamesJ
    • 4 years ago
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    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.

  2. sswann222
    • 4 years ago
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    using that matrix i have to do row reduction methods?

  3. JamesJ
    • 4 years ago
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    In which case, show that this matrix can be row reduced to the identity matrix. That's an equivalent statement.

  4. sswann222
    • 4 years ago
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    I can get it to row reduced to the idenity which means that all three are linearly independent?....correct

  5. JamesJ
    • 4 years ago
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    Yes.

  6. sswann222
    • 4 years ago
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    thank you

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