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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.

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closed as off-topic by Sanchayan Dutta Jan 7 at 15:14

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    $\begingroup$ I would think these kinds of questions would go to math.stackexchange instead. They fit within the context of quantum computing so don't merit that flag, but they are applicable in general so I think this should be up for meta discussion. $\endgroup$ – AHusain Jun 21 '18 at 3:59
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    $\begingroup$ Please see quantumcomputing.meta.stackexchange.com/q/250/391 :). Although, it is a completely valid comment. There are a whole swathe of questions in the earlier parts of N&C on classical circuits and linear algebra that, while contextually relevant, are not strictly to do with quantum computation. This was supposed to be the first in the series of questions from the book, but perhaps we should start from a later chapter. $\endgroup$ – SLesslyTall Jun 21 '18 at 7:40
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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.

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