Jamarcus Lindsey

2022-10-08

I came across a problem while proving a very simple statement.

We have three vectors $T=\{{v}_{1},{v}_{2},{v}_{3}\}$, and the set of their sums $S=\{{v}_{1}+{v}_{2},{v}_{1}+{v}_{3},{v}_{2}+{v}_{3}\}$ for ${v}_{i}\in V$, for V any vector space. I have to prove that for rational coefficients $a,b,c\in \mathbb{Q}$, the following statement is true:

T linearly independent $\iff $ S linearly independent

$\Rightarrow $ was quite easy and I had no problem calculating it.

However, $\Leftarrow $ is where I faced the problem and I would appreciate any help and besides that, does $\Leftarrow $ is true for real coefficients.

We have three vectors $T=\{{v}_{1},{v}_{2},{v}_{3}\}$, and the set of their sums $S=\{{v}_{1}+{v}_{2},{v}_{1}+{v}_{3},{v}_{2}+{v}_{3}\}$ for ${v}_{i}\in V$, for V any vector space. I have to prove that for rational coefficients $a,b,c\in \mathbb{Q}$, the following statement is true:

T linearly independent $\iff $ S linearly independent

$\Rightarrow $ was quite easy and I had no problem calculating it.

However, $\Leftarrow $ is where I faced the problem and I would appreciate any help and besides that, does $\Leftarrow $ is true for real coefficients.

Mario Monroe

Beginner2022-10-09Added 12 answers

Note that the proof of $\Leftarrow $ is the same as that of $\Rightarrow $ only that

$T={\textstyle \{}\frac{1}{2}({w}_{1}+{w}_{2}-{w}_{3}),\frac{1}{2}({w}_{1}+{w}_{3}-{w}_{2}),\frac{1}{2}({w}_{2}+{w}_{3}-{w}_{1}){\textstyle \}}$

where $S=\{{w}_{1},{w}_{2},{w}_{3}\}$

$T={\textstyle \{}\frac{1}{2}({w}_{1}+{w}_{2}-{w}_{3}),\frac{1}{2}({w}_{1}+{w}_{3}-{w}_{2}),\frac{1}{2}({w}_{2}+{w}_{3}-{w}_{1}){\textstyle \}}$

where $S=\{{w}_{1},{w}_{2},{w}_{3}\}$

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