Prove that v2 is the only element in W_1\cap W_2.

[transgalactic]

V is a vector space on field F and there is a seriesv=(v_1,v_2,v_3,v_4)
which is independent on V
W_1=sp(v1,v2)
W_2=sp(v2,v3)
of V
prove that
<br /> W_1\cap W_2=sp(v2) <br />
its obvious v2 exists in both groups .
how am i supposed to prove it
??

 

[HallsofIvy]

V is a vector space on field F and there is a seriesv=(v_1,v_2,v_3,v_4)
which is independent on V
W_1=sp(v1,v2)
W_2=sp(v2,v3)
of V
prove that
<br /> W_1\cap W_2=sp(v2) <br />
its obvious v2 exists in both groups .
how am i supposed to prove it
??

Since
W_1\cap W_2 is a subspace of W₁, and W₁ has dimension 2, it has dimension 1 or 2. If it has dimension 2, the it is equal to W₂. But W_1 contains v1 which is not in W_1\cap W_2 so it is not of dimension 2.