Find a basis for the subspace of $\mathbb{R}^3$ determined by the line
$x=-3t .\ y=2t ,\ z=t$.
It seems to me that a basis for this subspace would be simply $\{t(-3,2,1)\}$, but could it really be so basic?
But I believe I understand how to find a basis for a solution space for a homogeneous system, which is the previous problem, and this seems of a similar nature, but it seems so much more involved than this so as to lead me to believe that it couldn't be so simple an answer as this.
Perhaps I am misunderstood?
Best Answer
Yes, it is that basic. However, a basis is given by, for instance, $\{(-3,2,1)\}$. Remember that a basis consists of a single vector in this case (the space is one-dimensional), so $t$ has no business appearing in your final answer. What you've written is a full description of all the vectors in the subspace.