GO Classes CS/DA 2025 | Weekly Quiz 3 | Fundamental Course and Linear Algebra | Question: 14

Answer 1

Option A— False

A given set of vectors is linearly independent if and only if none of the vectors can be expressed as linear combination of other vectors.

Option B – True

This follows the definition of Linear Dependence
A set of vectors are linearly dependent if atleast one vector can be expressed as the linear combination of other vectors

Option C – False

Counter example u = [0,1]; v=[1,0]; w= [2,3]
Here u cannot be expressed as linear combination of v,w. Only w can be w=2v+3u

Option D – False

Counter example:  u = [0,1]; v=[1,0]; w= [2,3]
{u,v} and {v,w} are linearly dependent but not {u,v,w} as w can be expressed as w=2v+3u

Answer 2

in option A it is wrong beacuse if u is not Linear combination of v,w maybe v=C*w then {u,v,w} are LD

in option B it is correct beacuse by the definition of LD  atleast one vector is Linear combination of other

in option c it say if {u,v,w} are LD then only possibility is u is Linear combination of v and w but maybe it is Linearly dependent because of v=c*w

in option d it is wrong take this counter example u={1,0},v={0,1},w={5,6}

here u,v->LI v,w->LI

still it is Linearly dependent

Answer : B