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?

Contributor

David Cherney, Tom Denton, and Andrew Waldron (UC Davis)