Math Help - Vector expressed in function of two other vectors.

1. Vector expressed in function of two other vectors.

Hi!

This is for a homework I had to submit yesterday. I think I managed to answer all the questions correctly, except the last one.

I had the following U set and v vector:

$U = [8\vec{i} - \vec{j}, 6\vec{i} - \vec{k}]$
$\vec{v} = \vec{i} + 4\vec{j} + 9\vec{k}$

I had to prove that U is a subspace for V³, which I did.

Then I found:

$
U = \{\vec{u} \in V^3 \, | \, \vec{u} = x\vec{i} + y\vec{j} + z\vec{k} \; where \, -x - 8y + 6z = 0 \}
$

Then I was asked to find a base for this set, and I found:

$B = (-8\vec{i} + \vec{j}, 6\vec{i} + \vec{k})$

but I could also just have used the U specified above as a base. (I think)

I was then asked to find a base for $U^{\top}$, I got:

$
U^{\top} = \{\vec{w} \in V^3 \, | \, \vec{w} = x\vec{i} + y\vec{j} + z\vec{k} \; where \, 14x - y - z = 0 \}
$

and:

$B = (-8\vec{i} + \vec{j}, 6\vec{i} + \vec{k})$

found previously also belongs to $U^{\top}$.

Finally, I was asked to say if $\vec{v}$ belonged to U, and if not to express $\vec{v}$ in function of $\vec{u}$ and $\vec{w}$:

I plugged the values of v in $-x - 8y + 6z$ and it gave me 21, which is not equal to 0, so I said it wasn't in U. That's when my problems started, I was never able to express $\vec{v}$ in function of $\vec{u}$ and $\vec{w}$!

I tried:

$\vec{v} = m\vec{u} + n\vec{w}$
$\vec{i} + 4\vec{j} + 9\vec{k} = m(-2\vec{i} + \vec{j} + \vec{k}) + n(-\vec{i} - 8\vec{j} + 6\vec{k})$

Which gave me 3 equations and I tried to solve for m and n, but I wasn't able to. Now I know I have two orthogonal vectors (so they form a plane) and I know I need to use them to describe a third vector, so this third vector HAS to be on that plane too (that's why it doesn't work above). The problem I'm stuck with is that I don't know how to find $\vec{u} \in U$ and $\vec{w} \in U^{\top}$ that also form a plane in which $\vec{v}$ is!

Thanks in advance for helping me out.

2. My exam about this is this Friday, so I would love to have some insight before it.

Thanks a lot again!