Question

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.

Answer 1

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.

Answer 2

using that matrix i have to do row reduction methods?

Answer 3

In which case, show that this matrix can be row reduced to the identity matrix. That's an equivalent statement.

Answer 4

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

Answer 5

Yes.

Answer 6

thank you