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Reproduced from Exercise 2.1 of Nielsen & Chuang's Quantum Computation and Quantum Information (10th Anniversary Edition):
Show that $(1, −1)$, $(1, 2)$ and $(2, 1)$ are linearly dependent.
Note: This question is part of a series attempting to provide worked solutions to the exercises provided in the above book.
A set of $n$ vectors $V = \{\vec{v}_1, \ldots, \vec{v}_n\}$ are linearly dependant if there exists a set of scalars $a_1, \ldots, a_n$ (which are not all zero) such that $$ \sum_{i=1}^n a_i\vec{v_i} = \vec{0} $$ where $\vec{0}$ is the all-zero vector.
Writing $V$ as a matrix with vectors as columns, this is equivalent to finding a solution to the matrix equation $$ V\vec{a} = \vec{0}, \quad\textrm{where}\quad \vec{a} = \begin{pmatrix} a_1\\ \vdots \\ a_n \end{pmatrix}. $$
In our case, we have $$ V = \begin{pmatrix} 1 & 1 & 2 \\ -1 & 2 & 1 \end{pmatrix}, $$ with a solution to $V\vec{a} = \vec{0}$ provided by $$ \vec{a} = \begin{pmatrix} 1\\ 1 \\ -1 \end{pmatrix}, $$ hence showing linear dependance.
