Let w be a vector in the vector space V. Show that the sets of vectors {w, 0} and {w, -w} are linearly dependent.

Question

Let w be a vector in the vector space V.  Show that the sets of vectors {w, 0} and {w, -w} are linearly dependent.

turingtest · Answer

Just take the determinant of the matrix formed with those vectors as columns. If it's non-zero the set is linearly independent

anonymous · Answer

But it doesn't say how many entries each vector has or anything, it just says a vector w of anysize.

turingtest · Answer

you mean the vectors are not$$\langle w,0\rangle$$and$$\langle w,-w\rangle$$?

turingtest · Answer

that represents something else?

anonymous · Answer

i take it as $$w=\left(\begin{matrix} a _{1}\ ..\a _{n}\ \end{matrix}\right)$$
and the first is the set of the two vectors w and the 0 vectors, and the next is the set w and -w... not as w being a single value entry

turingtest · Answer

That really doesn't make a lot of sense to me, I must say. I don't see much logic in saying that the set$$\{\vec w,\vec 0\}$$is linearly independent. I could be completely misreading the question though...

turingtest · Answer

I also don't understand what it means to compare the two sets like that if that';s what they want.

anonymous · Answer

Can I just say... if A={w,0}, and then since Ax=0 that not just the trivial solution exists for (x,y), and y can be any value, that the systems is dependent.

I believe they are saying, show that the two vectors w, and the zero vector are dependent... and also show that the two vectors w and -w are dependent... showing two separate cases.