91Ó°ÊÓ

$\text{n}$-dimensional space is given.

If a nontrivial linear combination of vectors equals 0, a set of vectors is linearly dependent.

We assume that the n-dimensional space has a normalized set of bases $\left(e_1,e_2,……….,e_n\right)$, which is the standard basis for this space. Any vector in this

space is $V_k=\underset{i}{\overset{n}{∑}}{C}_i{e}_i$; and in this case, k runs from $1→n+1$i.e., we have $(\text{n}+1)$vectors. The first unit vector can be calculated as shown below.

$e_1=\frac{1}{C_1}(V_1-\underset{i=2}{\overset{n}{∑}})$

All the components of the other unit vectors were subtracted from $V_1$, except that of the first unit vector. Then they were divided by the corresponding coefficient.

From this we conclude that the $n$-dimensional space is spanned by the set $\left(V_1,e_2,………,e_n\right)$.

If we continue this way for the remaining vectors up to vector $V_n$, we find that the $n$ dimensional space is spanned by the set $\left(V_1,V_2,………,V_n\right)$ and hence the vectors $V_{n+1}$ is just a linear combination of the other $n$ vectors which make the set of $(n+1)$ vectors a linearly dependent set.