<s>[INST] Determine whether this set of vectors is linearly independent For $n \geq 1$ determine if the set $S$ is linearly independent.
The set $S$ is these vectors: 
$$
v_1 \equiv(\,1,2,\ldots,n\,)\,,\quad
v_2 \equiv(\,1,2^2,3^2,\ldots,n^2\,)\,,\ldots\,
v_n \equiv(\,1,2^n,3^n,\ldots,n^n\,)
$$
I know for vectors to be linearly independent we usually multiply by a scalar  to each vector and see if it equals the zero vector
(i.e. $av_1 + bv_2 + cv_3=0$ ), but I've attempted it that way but am having trouble solving it when its a series involving $n$'s.
All help would be appreciated
 [/INST] If $n=3$ it is
$$\left|\begin{array}{rrr} 1 & 1 & 1 \\ 2 & 2^2 & 2^3 \\ 3 & 3^2 &  3^3 \end{array}\right|=\left|\begin{array}{rrr} 1 & 1 & 1 \\ 0 & 2 & 2^2 \\ 0 & 2\cdot 3 &  2\cdot 3^2 \end{array}\right|=\left|\begin{array}{rr}   2 & 2^2 \\  2\cdot 3 &  2\cdot 3^2 \end{array}\right|=2^2\cdot 3 \left|\begin{array}{rr}   1 & 2 \\  1 &  3 \end{array}\right|\ne 0.$$
If $n=4$ it is
$$\left|\begin{array}{rrrr}1 & 1 & 1 & 1 \\ 2 & 2^2 & 2^3 & 2^4 \\ 3 & 3^2 & 3^3 & 3^4 \\4 & 4^2 & 4^3 & 4^4 \end{array}\right|=\left|\begin{array}{rrrr}1 & 1 & 1 & 1 \\ 0 & 2 & 2^2 & 2^3 \\ 0 & 2\cdot 3 & 2\cdot 3^2 & 2\cdot 3^3 \\ 0& 3\cdot 4 & 3\cdot 4^2 & 3\cdot 4^3 \end{array}\right|=\left|\begin{array}{rrr} 2 & 2^2 & 2^3 \\ 2\cdot 3 & 2\cdot 3^2 & 2\cdot 3^3 \\ 3\cdot 4 & 3\cdot 4^2 & 3\cdot 4^3 \end{array}\right| \\=2^2\cdot 3^2 \cdot 4\left|\begin{array}{rrr} 1 & 2 & 2^2 \\ 1 & \cdot 3 & 3^2 \\ 1 & 4 &  4^2 \end{array}\right|\ne 0.$$
You can get the solution for arbitrary $n$ using induction.
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