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

  1. #1
    Junior Member
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    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 V, which I did.

    Then I found:

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

    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:

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

    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.
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  2. #2
    Junior Member
    Joined
    Feb 2009
    Posts
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    My exam about this is this Friday, so I would love to have some insight before it.

    Thanks a lot again!
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