# Nielsen & Chuang Exercise 2.1 - “Linear dependence: example” [closed]

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.

## closed as off-topic by Sanchayan DuttaJan 7 at 15:14

• This question does not appear to be about quantum computing or quantum information, within the scope defined in the help center.
If this question can be reworded to fit the rules in the help center, please edit the question.

## locked by Sanchayan DuttaJan 7 at 15:17

This question exists because it has historical significance, but it is not considered a good, on-topic question for this site so please do not use it as evidence that you can ask similar questions here. This question and its answers are frozen and cannot be changed. See the help center for guidance on writing a good question.

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.