91Ó°ÊÓ

Consider a basis $\left\{f_1,f_2,f_3\right\}$of V and a set $\{f_1,f_1+f_2,f_1+f_2+f_3\}$.

The set $\{1,t,t_2,…,t_n\}$is linear independent set of V if there exist constant${\mathbf{a}}_{\mathbf{i}}\mathbf{∈}\mathbf{R}$such that${\mathbf{a}}_{\mathbf{0}}\mathbf{+}{\mathbf{a}}_{\mathbf{1}}{\mathbf{t}}_{\mathbf{1}}\mathbf{+}{\mathbf{a}}_{\mathbf{2}}{\mathbf{t}}_{\mathbf{2}}\mathbf{+}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{+}{\mathbf{a}}_{\mathbf{n}}{\mathbf{t}}_{\mathbf{n}}\mathbf{=}\mathbf{0}$where ${\mathbf{a}}_{\mathbf{0}}\mathbf{}\mathbf{=}{\mathbf{a}}_{\mathbf{1}}\mathbf{=}{\mathbf{a}}_{\mathbf{2}}\mathbf{=}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{=}{\mathbf{a}}_{\mathbf{n}}\mathbf{=}\mathbf{0}$.

Consider the constant$b_i$such that$b_1{f}_1+b_2{f}_2+b_3{f}_3=0$implies $b_1=b_2=b_3=0$.

Assume$a_i∈R$such that $a_1{f}_1+a_2\left(f_1+f_2\right)+a_3(f_1+f_2+f_3)=0$.

Simplify the equation$a_1{f}_1+a_2\left(f_1+f_2\right)+a_3(f_1+f_2+f_3)=0$as follows.

localid="1659438180898" $$\begin{gathered} \begin{array}{ll}{a}_1{f}_1+a_2\left(f_1+f_2\right)+a_3(f_1+f_2+f_3)=0& \end{array} \\ {a}_1{f}_1+a_2{f}_1+a_2{f}_2+a_3{f}_1+a_3{f}_2+a_3{f}_3=0 \\ \left(a_1+a_2+a_3\right)f_1+(a_2+a_3)f_2+a_3{f}_3=0 \end{gathered}$$

Compare the equation$\left(a_1+a_2+a_3\right)f_1+(a_2+a_3)f_2+a_3{f}_3=0$both side as follows.

localid="1659437781692" $\begin{array}{l}\left(a_1+a_2+a_3\right)f_1+(a_2+a_3)f_2+a_3{f}_3=\left(0\right)f_1+\left(0\right)f_2+\left(0\right)f_3\\ a_1+a_2+g_3=0\\ a_2+a_3=0\\ a_3=0\end{array}$

Substitute the values 0 for$a_3$in the equation$a_2+a_3=0$as follows.

$\begin{array}{l}{a}_2+a_3=0\\ a_2+\left(0\right)=0\\ g_2=0\end{array}$

Substitute the values 0 for $a_3$and 0 for$a_2$ in the equation localid="1659438539351" $a_1+a_2+a_3=0$as follows.

$\begin{array}{l}{a}_1+a_2+a_3=0\\ a_1+\left(0\right)+\left(0\right)=0\\ e_1=0\end{array}$

As $a_i=0$, by the definition of linear independence the subset$\{f_1,f_1+f_2,f_1+f_2+f_3)$is L.I.

Hence, the statement is True.