Exercise 2.1#
To check whether the given three vectors \((1,-1), (1,2)\) and \((2,1)\) are linearly dependent, we could search for the solution of following equation
(1)#\[\begin{split}
\begin{pmatrix}
1 & 1 & 2\\
-1 & 2 & 1\\
\end{pmatrix}\begin{pmatrix}
a \\ b \\ c
\end{pmatrix} = \begin{pmatrix} 0\\0 \end{pmatrix} \iff \begin{cases}
a + b + 2c = 0 \\
-a + 2b + c = 0
\end{cases}
\end{split}\]
One of the valid solution is \(a = 1, b = 1, c = -1\). Then we could conclude that the given three vectors \((1,-1), (1,2)\) and \((2,1)\) are linearly dependent.