In attempting to learn linear algebra I'm working through Strang's "Introduction to Linear Algebra", 4th ed. and (embarrassingly) am stuck on the first problem.
The problem is:
Describe Geometrically (line, plane, or all of $\mathbf{R}^3$) all linear combinations of:
\begin{align} \begin{bmatrix} 2 \\ 0 \\ 0 \end{bmatrix}\text{ and } \begin{bmatrix} 0 \\ 2 \\ 2 \end{bmatrix}\text{ and } \begin{bmatrix} 2 \\ 2 \\ 2 \end{bmatrix} \end{align}
To me, it seems that since the third vector is the sum of the first two, this should fill a plane inside of $\textbf{R}^3$ (specifically, containing all points for which $y=z$).
However, in the back of the book, the answer is "all of $\textbf{R}^3$". I must be missing something, but how can this be? What combination of these vectors produces the point $(x=0,y=1,z=2)$ for example?
Given that (1) this is the first problem in the book and that (2) I couldn't find any errata mentioning this, and (3) that I am a noob, I am inclined to think the problem is my understanding rather than a mistake in the book.
For sake of having an answer, note that the author is wrong: any vector $(x,y,z)$ with $y\ne z$ is clearly not a linear combination of the three mentioned vectors.