TITLE: Check if polynomials are linearly independent.
QUESTION [7 upvotes]: I would like to check if polynomials $1, 1+t^2, 1+t+t^2$ are linearly independent.
My idea is:
$1 \to [1,0,0]$
$1+t^2\to [1,1,0]$
$1+t^2+t^3 \to [1,1,1]$
And now $\left( \begin{array}{ccc}
1 & 0 & 1 \\
0 & 1 & 1 \\
0 & 1 & 1 \end{array} \right)$
I would like to find rank of this array. Rank of this array is $3$ so columns are linearly independent.   
Is it correct reasoning ?

REPLY [11 votes]: Just for fun you could use the definition of linear independence. That is, say that
$$
a(1) + b(1 + t^2) + c(1 + t + t^2) = 0.
$$
Then
$$
(a + b + c)1 + (c)t + (b + c)t^2 = 0. 
$$
Now you probably know that $\{1,t,t^2\}$ is linearly independent. So
$$
\begin{align}
c &= 0 \\
b + c &= 0 \Rightarrow b = 0 \\
a + b + c &= 0 \Rightarrow a = 0.
\end{align}
$$