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Mathematics LibreTexts

10.4: Review Problems

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    1. Let \(B^{n}\) be the space of \(n\times 1\) bit-valued matrices (\(\textit{i.e.}\), column vectors) over the field \(\mathbb{Z}_{2}\).
    Remember that this means that the coefficients in any linear combination can be only \(0\) or \(1\), with rules for adding and multiplying coefficients given here.

    a) How many different vectors are there in \(B^{n}\)?

    b) Find a collection \(S\) of vectors that span \(B^{3}\) and are linearly independent. In other words, find a basis of \(B^{3}\).

    c) Write each other vector in \(B^{3}\) as a linear combination of the vectors in the set \(S\) that you chose.

    d) Would it be possible to span \(B^{3}\) with only two vectors?

    2. Let \(e_{i}\) be the vector in \(\Re^{n}\) with a \(1\) in the \(i\)th position and \(0\)'s in every other position. Let \(v\) be an arbitrary vector in \(\Re^{n}\).

    a) Show that the collection \(\{e_{1}, \ldots, e_{n} \}\) is linearly independent.

    b) Demonstrate that \(v=\sum_{i=1}^{n} (v\cdot e_{i})e_{i}\).

    c) The \(span \{e_{1}, \ldots, e_{n} \}\) is the same as what vector space?

    3. Consider the ordered set of vectors from \(\Re^{3}\)
    $$
    \left( \begin{pmatrix}1\\2\\3\end{pmatrix} , \begin{pmatrix}2\\4\\6\end{pmatrix}, \begin{pmatrix}1\\0\\1\end{pmatrix} , \begin{pmatrix}1\\4\\5\end{pmatrix} \right)
    \]

    a) Determine if the set is linearly independent by using the vectors as the columns of a matrix \(M\) and finding \({\rm RREF}(M)\).

    b) If possible, write each vector as a linear combination of the preceding ones.

    c) Remove the vectors which can be expressed as linear combinations of the preceding vectors to form a linearly independent ordered set. (Every vector in your set set should be from the given set.)

    4. Gaussian elimination is a useful tool figure out whether a set of vectors spans a vector space and if they are linearly independent.
    Consider a matrix \(M\) made from an ordered set of column vectors \((v_{1},v_{2},\ldots,v_{m})\subset \mathbb{R}^{n}\) and the three cases listed below:

    a) \({\rm RREF}(M)\) is the identity matrix.

    b) \({\rm RREF}(M)\) has a row of zeros.

    c) Neither case i or ii apply.

    First give an explicit example for each case, state whether the column vectors you use are linearly independent or spanning in each case. Then, in general, determine whether \((v_{1},v_{2},\ldots,v_{m})\) are linearly independent and/or spanning \(\mathbb{R}^{n}\) in each of the three cases. If they are linearly dependent, does \({\rm RREF}(M)\) tell you which vectors could be removed to yield an independent set of vectors?

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