Question: Let \(B^{n}\) be the space of \(n \times 1\) bit-valued matrices (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.

1. How many different vectors are there in \(B^{n}\) ?
2. Find a collection \(S\) of vectors that \(\operatorname{span} B^{3}\) and are linearly independent. In other words, find a basis of \(B^{3}\).
3. 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?

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