to prove the statement " if {v1, v2, v3, v4} is linearly independent, then {v1, v2, v3} is also linearly independent." it is easier to prove the contrapositive: "if {v1, v2, v3} is linearly dependent, then {v1, v2, v3, v4} is also linearly dependent."
Prove the contrapositive.

Question

to prove the statement " if {v1, v2, v3, v4} is linearly independent, then {v1, v2, v3} is also linearly independent." it is easier to prove the contrapositive: "if {v1, v2, v3} is linearly dependent, then {v1, v2, v3, v4} is also linearly dependent."
Prove the contrapositive.

anonymous · Answer

Have any thoughts?

helder_edwin · Answer

by definition if $\{v_1,v_2,v_3\}$ is linearly dependent then there are scalars $a,b,c$ not all zero such that
$$ \large av_1+bv_2+cv_3=0 $$
from this we get
$$ \large av_1+bv_2+cv_3+0v_4=0 $$
where not all the scalars are zero. I.e., the set $\{v_1,v_2,v_3,v_4\}$ is linearly dependent.

anonymous · Answer

@Jemurray3 , i do actually. i know that in order for a set to be linearly dependent, at least one of the vectors have to be linear combinations of the others. i think the question is telling us that  {v1, v2, v3} are linear combinations of each other. (not really sure though) and then i get stuck..

anonymous · Answer

As mentioned above, the (equivalent) definition of linearly dependent vectors is a set such that the only combination of them that equals zero has zero for all its coefficients.

helder_edwin · Answer

@Jemurray3 
u meant "independent" not "dependent". right?

anonymous · Answer

yep