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.