Independent Subspace: Proving (or Disproving) Linear Independence

[hayu601]

Suppose B = {b1,...,bn} and C={c1,...,cn} both are basis set for space V.
D = {d1,...,dn} is basis for space T.

If B and D is linearly independent, is C and D always independent too? How can we prove (disprove) it?

 

[HallsofIvy]

I don't know what you mean by two sets of vectors being "independent". By saying that "B and D is linearly independent" do you mean that the set $B\times D$ is a set of independent vectors in $V\times T$?

 

[hayu601]

It means that every bi element B is not linear combination of vectors in D

 

[Chantry]

If B and C are both separate basis for V, then C = aB.

And if B and D are linearly independent, Bb != D and thus, Bab = Cb != Da

So Cd != D for some scalar d=b/a

That's basically what you have to prove in a more elegant form.