Thread: Linearly independent set that spans same subspace as original set of vectors

1. Linearly independent set that spans same subspace as original set of vectors

Problem:

Determine a linearly independent set of vectors that spans the same subspace of V as that spanned by the original set of vectors.

V=R3, {(1,1,1),(1,-1,1),(1,-3,1),(3,1,2)}

Question:

There is a theorem that says, "If one of the vectors in the set is a linear combination of the other vectors in the set, then that vector can be deleted from the given set of vectors".

I found the following dependency relationship among these vectors to be:

v1-2v2+v3=0

Now, I can solve for any one of the vectors in terms of the other two. Does this mean I can remove one, two, or three vectors from the original set, or am I limited to just one "vector removal"?

Thanks for any help!

2. Originally Posted by divinelogos
Problem:

Determine a linearly independent set of vectors that spans the same subspace of V as that spanned by the original set of vectors.

V=R3, {(1,1,1),(1,-1,1),(1,-3,1),(3,1,2)}

Question:

There is a theorem that says, "If one of the vectors in the set is a linear combination of the other vectors in the set, then that vector can be deleted from the given set of vectors".

I found the following dependency relationship among these vectors to be:

v1-2v2+v3=0

Now, I can solve for any one of the vectors in terms of the other two. Does this mean I can remove one, two, or three vectors from the original set, or am I limited to just one "vector removal"?

Thanks for any help!
$\begin{vmatrix} 1 & 1 & 1 \\ 1& -1 & 1 \\ 3& 1 & 2\end{vmatrix}=2$

So these vectors are linearly independent so they span all of $\mathbb{R}^3$

Since they only have to span the same space why not just pick the standard basis vectors.

3. Originally Posted by TheEmptySet
$\begin{vmatrix} 1 & 1 & 1 \\ 1& -1 & 1 \\ 3& 1 & 2\end{vmatrix}=2$

So these vectors are linearly independent so they span all of $\mathbb{R}^3$

Since they only have to span the same space why not just pick the standard basis vectors.
Can you elaborate a little? By standard basis vectors do you mean multiples of (1,0,0) (0,1,0) and (0,0,1)?

4. Originally Posted by divinelogos
Can you elaborate a little? By standard basis vectors do you mean multiples of (1,0,0) (0,1,0) and (0,0,1)?
That's correct.