Let-vector-set-u-1-u-2-u-3-u-4-in-C-n-free-linear-So-that-u-1-u-2-u-2-u-3-u-3-u-4-u-4-u-1-too-free-linear-scalar-have-to-

Question Number 54955 by gunawan last updated on 15/Feb/19

Let vector set {u_{1}, u_{2}, u_{3}, u_{4}} in C^{n}

free linear . So that

{u_{1} + α u_{2}, u_{2} + α u_{3}, u_{3} + α u_{4}, u_{4} + α u_{1}}

too free linear, scalar α have to \dots

Answered by kaivan.ahmadi last updated on 15/Feb/19

c_{1} (u_{1} + α u_{2}) + c_{2} (u_{2} + α u_{3}) + c_{3} (u_{3} + α u_{4}) + c_{4} (u_{4} + α u_{1}) = 0 \Rightarrow

(c_{1} + c_{4} α) u_{1} + (c_{1} α + c_{2}) u_{2} + (c_{2} α + c_{3}) u_{3} + (c_{3} α + c_{4}) u_{4} = 0 \Rightarrow s i n c e u_{i} a r e f r e e l i n e a r

{\begin{cases} c_{1} + c_{4} α = 0 \Rightarrow c_{1} = - c_{4} α \\ c_{1} α + c_{2} = 0 \Rightarrow c_{4} α^{2} = c_{2} \\ c_{2} α + c_{3} = 0 \Rightarrow - c_{4} α^{3} = c_{3} \\ c_{3} α + c_{4} = 0 \Rightarrow - c_{4} α^{4} + c_{4} = 0 \Rightarrow (1 - α^{4}) c_{4} = 0 \Rightarrow c_{4} = 0 \end{cases}

\Rightarrow c_{3} = c_{2} = c_{1} = 0

w e s h o u l d h a v e 1 - α^{4} \neq 0 \Rightarrow α^{4} \neq 1

Commented by gunawan last updated on 15/Feb/19

nice

thank you Sir

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