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Math Help - another basis and dimension question

  1. #1
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    another basis and dimension question

    Let the subset U \subset \mathbb{R}^4 be defined by

    U = \left \{ v | v = \left [\begin{matrix} v_1 \\ v_2 \\ v_3 \\ v_4 \end{matrix} \right ] , v_1 - v_2 + v_3 = 0 , v_2 + 2v_3 - v_4 = 0 \right \}

    1. Show that U is a sub-space of \mathbb{R}^4

    Because \mathbb{R}^4 is a vector space and U \subset \mathbb{R}^4 U inherits the same operations as \mathbb{R}^4. If U is a sub-space then

    a) U is non-empty, eg. \left [\begin{matrix} 0 \\ 0 \\ 0 \\ 0 \end{matrix} \right ] \in U

    b) p \in U and q \in U, then p + q \in U

    \left [\begin{matrix} p_4 - 3p_3 \\ p_4 - 2p_3 \\ p_3 \\ p_4 \end{matrix} \right ] + \left [\begin{matrix} q_4 - 3q_3 \\ q_4 - 2q_3 \\ q_3 \\ q_4 \end{matrix} \right ] = \left [\begin{matrix} (p_4 + q_4) - 3(p_3 + q_3) \\ (p_4 + q_4) - 2(p_3 + q_3) \\ (p_3 + q_3) \\ (p_4 + q_3) \end{matrix} \right ]

    c) \alpha \left [\begin{matrix} p_4 - 3p_3 \\ p_4 - 2p_3 \\ p_3 \\ p_4 \end{matrix} \right ] \in U

    2. Find a basis and dimension of U

    U = \left \{ v_4  \left [\begin{matrix} 1 \\ 1 \\ 0 \\ 1 \end{matrix} \right ] + v_3 \left [\begin{matrix} -1 \\ -2 \\ 1 \\ 0 \end{matrix} \right ] , v_3, v_4 \in \mathbb{R} \right \}

    pretty simple, just the two column vectors above and so, dim(U) is 2.

    3. How do I change the basis to a basis over \mathbb{R}^4 ?
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  2. #2
    Senior Member TheAbstractionist's Avatar
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    Hi bmp05.

    Express v_3 and v_4 in terms of v_1 and v_2.

    v_3\ =\ v_2-v_1

    v_4\ =\ 2v_3+v_2\ =\ 3v_2-2v_1

    Hence each vector in U is of the form

    \begin{bmatrix}v_1\\v_2\\v_2-v_1\\3v_2-2v_1\end{bmatrix}\ =\ v_1\begin{bmatrix}1\\0\\-1\\-2\end{bmatrix}\,+\,v_2\begin{bmatrix}0\\1\\1\\3\en  d{bmatrix}

    This shows that \left\{\begin{bmatrix}1\\0\\-1\\-2\end{bmatrix},\,\begin{bmatrix}0\\1\\1\\3\end{bma  trix}\right\} is a basis for U, which therefore has dimension 2. (Note that it also shows that U is a subspace of \mathbb R^4.)
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  3. #3
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    I believe the final question was how to extend the given basis, <br />
U = \left \{ v_4 \left [\begin{matrix} 1 \\ 1 \\ 0 \\ 1 \end{matrix} \right ] + v_3 \left [\begin{matrix} -1 \\ -2 \\ 1 \\ 0 \end{matrix} \right ] , v_3, v_4 \in \mathbb{R} \right \}<br />
to a basis for R^4. Just find two more independent vectors. That will give a set of 4 independent vectors which will form a basis for R^4. Looks to me like \left[\begin{matrix}1 \\ 0 \\ 0\\ 0\end{matrix}\right] and \left[\begin{matrix}0 \\ 1 \\ 0 \\ 0\end{matrix}\right] will do.
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