Chapter 2.4, Problem 9P

Explanation of Solution

Showing linear dependency of vectors:

Consider the following set of vectors,

$V=\left\{v_1,v_2,\dots ,v_n\right\}$

If this system of vectors is linearly dependent, then we have,

$c_1{v}_1+c_2{v}_2+\dots +c_n{v}_n=0$

For not all $c_i=0$.

Now for some vector $v_k$ in $V$ we can write as given below:

$c_1{v}_1+c_2{v}_2+\dots +c_k{v}_k+\dots +c_n{v}_n=0$

This can be written as follows:

$v_k=-\frac{1}{c_k}\left(c_1{v}_1+c_2{v}_2+\dots +c_{k-1}{v}_{k-1}+c_{k+1}{v}_{k+1}\dots +c_n{v}_n\right)$

Therefore, we can see that a vector can be written as non-trivial combination of other vectors