Geometrically Describing a Subspace I have a practice question here and it is asking to geometrically describe this subspace in $R^3$ It is asking if it is a point, a line, or a plane or all of three-space.
Here it is:
Span {(1,-2, 1) (-2, 4, -2)}
What I attempted was to show that it was linearly independent. I got that this is a plane, but I just need to clarify how to do this.
 A: Are you sure you have a plane? What happens if you multiply $(1,-2,1)$ by $2$?
In general, to solve these problems you need to prove or disprove linear independence, and if you can disprove it then you need to find a maximally linearly independent subset. The first part of the solution is always to set
$$a_1 v_1 + a_2 v_2 + \dots + a_n v_n = 0$$
where the $v_i$ are your set of vectors and the $a_i$ are unknown. If you can solve for nonzero $a_i$ as I just did then you don't have linear independence. The details are as follows:
$$v_1 = (1,-2,1), \ \ \ v_2 = (-2,4,-2), \ \ \ a_1 = 2, \ \ \ a _2 = 1.$$
Here, $n=2$, because there are $2$ vectors in your given set. Plugging into the first equation, we get
$$a_1 v_1 + a_2 v_2 = 2(1,-2,1) + 1(-2,4,-2) = (2 - 2, -4 + 4, 2 - 2) = (0,0,0),$$
so since a nonzero set of $a_i$ satisfy the equation, linear independence fails.
Furthermore, since one vector is always a linearly independent set as long as it's not the zero vector, either $\{(1,-2,1)\}$ or $\{(-2, 4, -2)\}$ is sufficient to form a maximally linear independent subset. This shows that the span of that set is $1$-dimensional--that is, a line.
