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If $X_1,X_2$ and $X_3$ are vectors in $\mathbb{R}^3$ such that $\{X_1,X_2\}$ and $\{X_1,X_3\}$ are linearly independent sets, then $\{X_1,X_2,X_3\}$ is a linearly independent set or not?

since linearly independent sets need to have consistent solution and no variable can be zero. so i am assuming that in $\{X_1,X_2,X_3\}$ , $X_2$ could make all variables zero making it linearly dependent? i am not sure it i am correct.

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Hint

For a counterexample, let $X_1, X_2$ be linearly independent vectors and take $X_{3}=2X_2$ for example.

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Guide:

  • Think of the situation when $x_2=x_3$. Is it linearly independent? If it is not obvious to you, try to work with concrete example.

  • What if $x_1, x_2, x_3$ arre the standard unit vectors?

Remark:

  • It is unclear to me what do you mean by no variable can be zero or $x_2$ can make all variables zero.
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Only because the two sets of $\{X_1,X_2\}$ and $\{X_1,X_3\}$ does not guarantees you that either $\{X_1,X_2,X_3\}$ or $\{X_2,X_3\}$ are linear indenpent. Just consider the three vectors

$$X_1~=~\begin{pmatrix}1\\2\\3\end{pmatrix} X_2~=~\begin{pmatrix}4\\5\\6\end{pmatrix} X_3~=~\begin{pmatrix}8\\10\\12\end{pmatrix}$$

As you can see you restrictions are fullfilled but neither the set $\{X_2,X_3\}$ nor the whole set $\{X_1,X_2,X_3\}$ are linear independent.

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