Q82: If $S = \{(2, 5), (3, 7), (4, -9), (-8,... - Mathematics - ICAR AIEEA PG 2024 - Statistical Sciences | Agrigyan

Linear dependence in vector spaces:
The vectors in $S$ are 2-dimensional vectors belonging to $\mathbb{R}^2$.
The dimension of $\mathbb{R}^2$ is 2 ($\dim(\mathbb{R}^2) = 2$).
By the fundamental dimension theorem of linear algebra, any set of more than $n$ vectors in an $n$-dimensional vector space must be linearly dependent.
Here, the set $S$ contains 4 vectors in $\mathbb{R}^2$. Since $4 \gt 2$, $S$ is necessarily linearly dependent.
Notice also that removing 1 vector leaves 3 vectors, which still exceeds 2, so the subsets in Options 3 and 4 are also linearly dependent.