Is $\{v_1,v_2,v_3\}$ a basis for $H$? I don't understand the answer to the exercise that I'm doing, can anyone explain it in a more clear manner?


  Let $ v_1 = \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}, v_2 = \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix}, v_3 = \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} $, and let $H$ be the set of vectors in $\Bbb R^3$
  whose second and third entries are equal. Then every vector in $H$ has
  a unique expansion as a linear combination of $v_1, v_2, v_3$, because
  
  $ \begin{bmatrix} s \\ t \\ t \end{bmatrix} =  s \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} + (t-s) \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix}  + s \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} $
  for any $s$ and $t$. Is $\{v_1,v_2,v_3\}$ a basis for $H$?
  
  Why or why not?


The answer given was:


  In order for the set to be a basis for $H$, $\{v_1,v_2,v_3\}$ must be a spanning set for $H$; that is, $H=\operatorname{Span}\{v_1,v_2,v_3\}$. The exercise shows that $H$ is a subset of $\operatorname{Span}\{v_1,v_2,v_3\}$, but there are vectors in $\operatorname{Span}\{v_1,v_2,v_3\}$ which are not in $H$ ($v_1$ and $v_3$, for example). So $H \neq \operatorname{Span}\{v_1,v_2,v_3\}$, and $\{v_1,v_2,v_3\}$ is not a basis for $H$.


I see $v_1$ and $v_3$ making $H$, why does it say they are not in $H$?
You'll notice that $H$ is indeed a subset of $\Bbb R^3$, but to be more precise, it's a subspace of it resembling $\Bbb R^2$ (it's a plane). To see this, notice that the first variable is free, and the third dependent upon the third.

Now, you can use the vectors $v_1, v_2, v_3$ to express points in that plane, but you'll notice that you can also use those vectors to express points that are not in the plane, $v_1$ for example, so $Span(v_1,v_2,v_3)$ does not span $H$.
you can express the standard basis $\{e_1, e_2, e_3\}$ as linear combinations of $\{v_1, v_2, v_3\}$ in the following way: $e_3 = \pmatrix{1\\0\\0} = v_2 - v_3, e_1 = \pmatrix{1\\0\\0}= v_1 - v_2 + v_3, e_2 = v_3.$  therefore $\{v_1, v_2, v_3\}$ spans $R^3.$  to show independence, suppose $$c_1v_1 + c_2 v_2 + c_3v_3 = 0$$ then you have $$\pmatrix{c_1\\c_2 + c_3\\c_1 + c_2} = \pmatrix{0\\0\\0}\to c_1 = 0, c_2 = 0, c_3 = 0. $$