## Exercises 2.7 Exercises and Practice Exam on Systems of Equations

ΒΆ*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.

- \(\S\) - Definition and Examples of Spaces (p.86): 30
- \(\S\) - Subspaces and Spanning Set (p.97): 43
- \(\S\) - Definitions and Examples of Independence (p.109): 35, Extra Credit for 43
- \(\S\) - Basis (p.118): 29
- \(\S\) - Dimension (p.125): 22, Extra Credit fo 33
- \(\S\) - Vector Spaces and Linear Systems (p.133): 29 and 30

*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.

- \(\S\) - Definition and Examples of Spaces (p.86): 17,18,19,22
- \(\S\) - Subspaces and Spanning Set (p.97): 20,22,25,26
- \(\S\) - Definitions and Examples of Independence (p.109): 20,21,22,32
- \(\S\) - Basis (p.118): 18,19,20,23,33,35
- \(\S\) - Dimension (p.125): 16,17,18
- \(\S\) - Vector Spaces and Linear Systems (p.133): 16-25

*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

and

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 \(H\text{?}\)

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

###### 3.

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

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

###### 4.

Find bases for \(Row\ A\) and \(Col A\) given:

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

then list three distinct elements of the space.

###### 7.

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

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

###### 8.

Show that

are linearly independent and write

as a linear combination of the two.

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:

###### 10.

Find the basis and dimension of the vector space: