linear Independence of three vector functions are the below vectors linearly independent?
$$\begin{bmatrix}1\\0\\1\\\end{bmatrix},\begin{bmatrix}t\\0\\t\\\end{bmatrix},and \space \begin{bmatrix}t^2\\0\\t^2\\\end{bmatrix}
$$
I am told they are, but am confused how to show that.  I thought the coefficients would be $c_1 = t^2$, $c_2 = t$, and $c_3 = 1$; but wouldn't that mean dependence instead of independence? 
 A: They are linearly independent. Assume that scalars $\alpha, \beta, \gamma \in \Bbb{R}$ satisfy 
$$\begin{bmatrix} 0 \\ 0 \\ 0\end{bmatrix} = \alpha\begin{bmatrix}1\\0\\1\end{bmatrix}+\beta\begin{bmatrix}t\\0\\t\end{bmatrix}+\gamma\begin{bmatrix}t^2\\0\\t^2\end{bmatrix} = \begin{bmatrix}\alpha + \beta t + \gamma t^2\\0\\\alpha + \beta t + \gamma t^2\end{bmatrix}$$
for all $t \in \Bbb{R}$. This is equivalent to $\alpha + \beta t + \gamma t^2 = 0, \forall t \in \Bbb{R}$. This quadratic polynomial is zero so all coefficients must be equal to $0$, meaning $\alpha = \beta = \gamma = 0$.
The point is that coefficients $\alpha, \beta, \gamma \in \Bbb{R}$ are constant, they don't depend on $t$.
A: Now, they are not linearly independent, since, for instance,$$t\begin{bmatrix}1\\0\\1\end{bmatrix}-\begin{bmatrix}t\\0\\t\end{bmatrix}+0\times\begin{bmatrix}t^2\\0\\t^2\end{bmatrix}=\begin{bmatrix}0\\0\\0\end{bmatrix}.$$
A: $$\begin{bmatrix}1\\0\\1\\\end{bmatrix},\begin{bmatrix}t\\0\\t\\\end{bmatrix} \space \begin{bmatrix}t^2\\0\\t^2\\\end{bmatrix}$$
You can show they are linearly independant on $\mathbb R$ by proving that:
$$c_1v_1+c_2v_2+c_3v_3=0 \implies c_1=c_2=c_3=0$$
$$ \begin{bmatrix}  1 & t &t^2 \\ 0 & 0 & 0 \\ 1 & t & t^2 \end{bmatrix} \begin{bmatrix} c_1 \\ c_2 \\ c_3 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$$
Or more simply ( which is true):
$$c_1+c_2t+c_3t^2=0 \implies c_1=c_2=c_3=0$$
Where the coefficients  $c_i$ are numbers not functions.
A: If there were c1 and c2 and c3 that made this vectors dependent, you would have that $c1+c2.t+c3.t^2=0$ for each t which means that c1,c2 and c3 are zero. You should consider c1 and c2 and c3 as cofficients.
