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A. $\{u_1,u_2,u_3\}$ could be a linearly dependent or linearly dependent set of vectors depending on the vector space chosen.

B. $\{u_1,u_2,u_3\}$ is a linearly dependent set of vectors unless one of $u_1,u_2,u_3$ is the zero vector.

C. $\{u_1,u_2,u_3\}$ is never a linearly dependent set of vectors.

D. $\{u_1,u_2,u_3,u_4\}$ could be a linearly dependent or linearly dependent set of vectors depending on the vectors chosen.

E. $\{u_1,u_2,u_3,u_4\}$ is always a linearly independent set of vectors.

F. $\{u_1,u_2,u_3,u_4\}$ is never a linearly independent set of vectors.

G. None of the above

I answered B but is incorrect.

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  • $\begingroup$ What do you mean by 'linearly dependent or linearly dependent'? Do you mean ''linearly dependent or linearly independent''? $\endgroup$ Jan 28, 2019 at 17:32
  • $\begingroup$ What I post is exactly the same as the question. It is "linearly dependent or linearly dependent". $\endgroup$ Jan 28, 2019 at 17:35
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    $\begingroup$ That makes no sense. It is probably a typo and meant 'linearly dependent or linearly independent' $\endgroup$ Jan 28, 2019 at 17:36
  • $\begingroup$ In $B$ too, it should be linearly independent, otherwise the wording is meaningless $\endgroup$ Jan 28, 2019 at 17:43

4 Answers 4

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You should give a bit more detail about your thoughts and efforts to solve this problem, so I won't give a full answer but just some hints:

  • The formulation of $A$ and $D$ is clearly wrong, it should be "linearly dependent or linearly independent", otherwise it does not make any sense.
  • There are two correct answers, one among $A,B,C$ and one among $D,E,F$
  • $u_1, u_2, u_3$ are any three vectors, you don't know anything about them
  • $u_4$, on the other hand, is a linear combination of $u_1, u_2$ and $u_3$. Try to write down what this means and to see if you can conclude between $D,E,F$.
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Take $u_1=(1,0,0),u_2=(0,1,0),u_3=(0,0,1),u_4=u_1+u_2+u_3=(1,1,1). u_4$ is a linear combination of $u_1,u_2,u_3$ as required, yet $\{(1,0,0),(0,1,0)(0,0,1)\}$ is a set of linearly independent vectors. So, $B$ is not true in general.

A counter-example to $C$ would be $u_1=(1,0),u_2=(0,1),u_3=u_4=u_1+u_2=(1,1)$. The set $\{u_1,u_2,u_3\}$ is dependent since $u_3$ is a linear combination of $u_1,u_2$.

$D,E$ are wrong right away; you know for a fact that $u_1,u_2,u_3,u_4$ are dependent as $u_4$ is given to be a linear combination of $u_1,u_2,u_3$. This also means that $F$ is true.

As far as $A$ is concerned, I'm inclined to say that it is not true. The linear dependence or independence of vectors does not rely on the choice of the vector space as long as the vectors lie in the vector space and the field is the same.

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I would also say that A is false. $u_1$, $u_2$ and so forth are vectors of a certain vector space. When defining a vector, you define it as an element of a certain vector space. There is no such thing as $u_1$ in different vector spaces.

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$B$ is clearly false: if one of the vectors is $0$, the set is dependent. So $B$ translates to: "the set is dependent unless dependent". But there is no reason for $\{u_1,u_2,u_3\}$ to be dependent. It could be either.

Similarly $A$ is basically nonsense, offering as it does two options which are the same.

Again, $D$ offers the same option twice. And it's not that it could be a dependent set: it is one.

So the correct option is $F$.

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