$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

# 10.4: Review Problems

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?