Question

Find a nontrivial relation among the following vectors:

$\left[\left.\begin{array}{c}1\\ 2\end{array}\right],\left[\begin{array}{c}2\\ 3\end{array}\right],\left[\begin{array}{c}3\\ 4\end{array}\right.\right]$ .

Short answer

A non-trivial relation between the required vectors is,$\left[\left.\begin{array}{c}3\\ 4\end{array}\right]=-\left[\begin{array}{c}1\\ 2\end{array}\right]+2\left|\begin{array}{c}2\\ 3\end{array}\right.\right]$

Step-by-step solution

Consider the set.

A subset W of the vector space ${\mathrm{ℝ}}^n$is called a (linear) subspace of ${\mathrm{ℝ}}^n$if it has the following three properties:

a. W contains the zero vector in ${\mathrm{ℝ}}^n$.

b. W is closed under addition: If ${\overrightarrow{w}}_1$and ${\overrightarrow{w}}_2$are both in W, then so is ${\overrightarrow{w}}_1+{\overrightarrow{w}}_2$.

c. W is closed under scalar multiplication: If $\overrightarrow{w}$is in W and k is an arbitrary scalar, then $k\overrightarrow{w}$is in W.

Let $v_1=\left[\left.\begin{array}{c}1\\ 2\end{array}\right],v_2=\left[\begin{array}{c}2\\ 3\end{array}\right],v_3=\left[\begin{array}{c}3\\ 4\end{array}\right.\right]$and $v_3=c_1{v}_1+c_2{v}_2$

Consider the matrix representation.

The matrix representation of the above vectors is,

$$\begin{gathered} v_3=c_1{v}_1+c_2{v}_2 \\ \Rightarrow \left[\begin{array}{c}3\\ 4\end{array}\right]=c_1\left[\begin{array}{c}1\\ 2\end{array}\right]+c_1\left[\begin{array}{c}2\\ 3\end{array}\right] \\ \Rightarrow \left[\begin{array}{c}{c}_1+2c_2\\ 2c_1+3c_2\end{array}\right]=\left[\begin{array}{c}3\\ 4\end{array}\right] \end{gathered}$$

Solve the above equations.

$c_1=-1,c_2=2$

Final answer.

A non-trivial relation between the required vectors is,$\left[\left.\begin{array}{c}3\\ 4\end{array}\right]=-\left[\begin{array}{c}1\\ 2\end{array}\right]+2\left|\begin{array}{c}2\\ 3\end{array}\right.\right]$