Describing the generated subpace

Question

I'm having trouble geometrically describing the subspace of $\mathbb{R}^3$ generated by these vectors: $$ (0,0,1), \ (0,1,1), \ (0,2,1).$$ I've tried to put it in a system, but that only led me to:

$$ x=0, \quad y=b+2c, \quad z=a+b+c. $$

So I'm not sure what to do next. Any help would be appreciated, thanks :)

Answer 1

This is correct. Your description seems okay to me. This is assuming $a, b, c$ are in your underlying field ($\Bbb R$). The subspace is $$\{(0, y, z)\mid y=b+2c, z=a+b+c\}.$$

Answer 2

$$a \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} + b \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} + c \begin{bmatrix} 0 \\ 2 \\ 1 \end{bmatrix} => \begin{bmatrix} 0 \\ b+2c \\ a+b+c \end{bmatrix} $$

$$ (a*0) + (b*0) + (c*0) = 0 $$ $$ (a*0) + (b*1) + (c*2) = b + 2c $$ $$ (a*1) + (b*1) + (c*1) = a + b + c $$

Answer 3

The other answers are algebraic descriptions, but the question asks for a geometric description. Note that your algebraic description tells you that one of these vectors is linearly dependent, and indeed this is obvious from looking because none have non-zero entries in the third component. So the geometric description you're looking for can be obtained from the algebraic system by noting that these vectors span a plane (and not the whole space!) through the origin (because they are a vector subspace) and that this plane contains any two of the three vectors in the list.

Answer 4

Since $$\text{rank}\begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 1 \\ 0 & 2 & 1 \end{pmatrix} = 2$$ the generated subspace has dimension $2$ (it is a plane), and it trivially is the plane given by points whose first coordinate is zero (vectors orthogonal to $(1,0,0)$).