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GO Classes CS/DA 2025 | Weekly Quiz 3 | Fundamental Course and Linear Algebra | Question: 9

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Given a set of vectors $\text{S}(|\text{S}| \geq n)$, with all vectors in $\mathbb{R}^n$, which of the following is a necessary and sufficient condition for the vectors of $\text{S}$ to be Linearly Dependent?

  1. Exactly $n$ vectors can be represented as a linear combination of other vectors of the set $\text{S}$.
  2. At least $n$ vectors can be represented as a linear combination of other vectors of the set $\text{S}$.
  3. At least one vector $u$ can be represented as a linear combination of any vector(s) of the set $\text{S}$.
  4. At least one vector $u$ can be represented as a linear combination of vectors (other than $u$ ) of the set $\text{S}$.
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This follows directly from the definition of Linear Independence. $\text{C}$ can't be the answer because it says "any vector(s)", which would include the vector $u$ itself. $\text{A} \;\&\; \text{B}$ are sufficient, but not necessary.
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We know that definitions are always “necessary and sufficient “ statement and it is implicit.So given a set of vectors S(|S|>=n) with all vectors in Rn is said to be linearly dependent if and only if at least one vector can be represented as the linear combination of the others means representing at least one vector of set S as the linear combination of the other vectors is the necessary and sufficient condition for S to be linearly dependent.option A and option B are the sufficient condition not the necessary ,and according to the definition C is neither necessary nor sufficient

And D is the correct option according to the definition.
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Given that the number of vectors in the set S are >=n.

2 cases can ply:

1. |S| (# vectors in set S) > n ------> It would mean that whatever be , as the no. of vectors in S is >n for R^n . It is always going to be linearly dependent.

2.  |S| (# vectors in set S)  =  n ------> If that is the case then we can say that if At least one vector can be represented as a linear combination of vectors (other than itself ) of the set S then we can say it to be Linearly dependent, else Linearly Independent.
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