## Exercises2.7Exercises and Practice Exam on Systems of Equations

Textbook Exercises:

These are the exercises from Hefferon's text which you need to hand in as part of your grade. The solutions to these are available in the Answers to Exercises supplement, but you need to, to the best of your ability, answer them in your own words.

Practice Exercises:

Here are some exercises from the Hefferon's text which you should try, the answers to all of these can be found in the Answers to Exercises supplement for Hefferon's text. You do not need to hand in these exercises.

Practice Exam:

In addition to the practice problems above below are the questions for your practice exam, these must be turned in when you come in for the exam. The practice exam counts for 5% of your exam grade and can earn you back 15% of any points you loose on the in-class portion of the exam.

###### 1.

Given a vector space

\begin{equation*} H=span\{\vec{v_1},\vec{v_2},\vec{v_3}\} \end{equation*}

and

\begin{equation*} -3\vec{v_1}+2\vec{v_2}+\vec{v_3}=\vec{0}, \end{equation*}

assuming that $\vec{v_i}\neq\vec{0}$ and that they are not multiples of one another, what is a possible basis for $H\text{?}$ What is the dimension of $H\text{?}$

Hint

In the definition of a basis what conditions must it satisfy?

###### 2.

Find $n$ such that $Row\ A$ is a subspace of $\mathbb{R}^n$ and find $m$ such that $Col\ A$ is a subspace of $\mathbb{R}^m$ given

\begin{equation*} A= \left[ \begin{array}{rrrr} 1 \amp 13 \amp 5 \amp 0 \\ 0 \amp 1 \amp 4 \amp 1 \end{array} \right]. \end{equation*}
###### 3.

Find a matrix $A$ such that the given set is $Col\ A\text{.}$

\begin{equation*} V= \left\{ \left[ \begin{array}{r} 3r-2s\\ s-4t\\ -r+4t\\ r+2s+t \end{array} \right]:r,s,t\in\mathbb{R} \right\} \end{equation*}
Hint

Rewrite the vectors described in the set as a linear combination of multiple vectors.

###### 4.

Find bases for $Row\ A$ and $Col A$ given:

\begin{equation*} A= \left[ \begin{array}{rrrr} 5 \amp 0 \amp 25 \amp 10 \\ 3 \amp 1 \amp 15 \amp 5 \\ 2 \amp -1 \amp 10 \amp 5 \end{array} \right]. \end{equation*}
Hint

You may need to row reduce the matrix to get an idea of what is happening.

###### 5.

Given a $7\times3$ matrix, what are the maximum and minimum dimensions for the row and column spaces?

###### 6.

Give a basis for the space

\begin{equation*} \mathcal{P}_3=\left\{a_0+a_1x+a_2x^2+a_3x^3\right\} \end{equation*}

then list three distinct elements of the space.

###### 7.

Is the given vector in the span of the given set?

\begin{equation*} \vec{v}= \left[ \begin{array}{r} 7\\ 3\\ 13 \end{array} \right] \mbox{ and } S= \left\{ \left[ \begin{array}{r} -1\\ 7\\ 0 \end{array} \right], \left[ \begin{array}{r} 1\\ 5\\ 0 \end{array} \right] \right\} \end{equation*}
Hint

You can try to look at the problem as a set of equations, but try not to over think it.

###### 8.

Show that

\begin{equation*} S= \left\{ \left[ \begin{array}{r} -1\\ 0\\ 2 \end{array} \right], \left[ \begin{array}{r} 1\\ 5\\ 0 \end{array} \right] \right\} \end{equation*}

are linearly independent and write

\begin{equation*} \vec{v}= \left[ \begin{array}{r} -2\\ 25\\ 14 \end{array} \right] \end{equation*}

as a linear combination of the two.

Hint

Think in terms of vector equations, but again stop and think before you start to calculate.

###### 9.

Find a basis for the solution set of this system:

\begin{align*} x_1+x_2+x_3+x_4\amp =0\\ -x_2+7x_3-4x_4\amp =0\\ x_1+8x_3-3x_4\amp =0 \end{align*}
###### 10.

Find the basis and dimension of the vector space:

\begin{equation*} V= \left\{ \left[ \begin{array}{r} r\\ s\\ t\\ w \end{array} \right]:r,s,t,w\in\mathbb{R}\mbox{ and }r+2t-w=0 \right\} \end{equation*}