How to find a linearly independent vector?

[UrbanXrisis]

Find a linearly independent set of vectors that spans the same subspace of R^3 as that spanned by the vectors

$$\left(\begin{array}{c} -2 & -1 & -2 \end{array}\right) , \left(\begin{array}{c} -2 & 3 & -8 \end{array}\right) , \left(\begin{array}{c} 0 & -2 & 3 \end{array}\right)$$


I'm not sure how to find a linearly independent vector. For a linearly dependency, the determinant of the matrix cannot equal zero. But how would i find two other 3x1 vectors that does not have linear dependency?

 

[devious_]

You could row reduce and look at the columns that correspond to the leading columns of your reduced matrix. Can you see why this would mean they're independent?

 

[pizzasky]

Since there is a zero (in the third vector) which simplifies calculations, you can just check to see if any of the three vectors can be formed as a linear combination of the other two. If yes, eliminate anyone of the vectors.
It will then become immediately clear if the two remaining vectors are linearly independent or not!
Having confirmed the linear independence of the two remaining vectors, the spanning property follows. You now have a set of basis vectors for this vector space!