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Elementary Linear Algebra: An Active Learning Approach

Section 5.6 Linear Independence, Span and Basis: Homework

Exercises Exercises

Exercise Group.

Determine if the set \(S\) of vectors is linearly independent. Explain how you know.
1.
\(S=\left\{ \left(\begin{array}{r} -3\\5 \end{array}\right), \left(\begin{array}{r} 6 \\ -10 \end{array}\right) \right\}\)
2.
\(S=\left\{ \left(\begin{array}{r} 2\\-3 \end{array}\right), \left(\begin{array}{r} 6 \\ 12 \end{array}\right) \right\}\)
3.
\(S=\left\{ \left(\begin{array}{r} -1\\2\\0 \end{array}\right), \left(\begin{array}{r} 4 \\ 2 \\ -2 \end{array}\right), \left(\begin{array}{r} 0 \\ 1 \\ 6 \end{array}\right) \right\}\)
4.
\(S=\left\{ \left(\begin{array}{r} -1\\2\\0 \end{array}\right), \left(\begin{array}{r} 4 \\ 2 \\ -2 \end{array}\right), \left(\begin{array}{r} 0 \\ 1 \\ 6 \end{array}\right), \left(\begin{array}{r} 3 \\ 1 \\ -1 \end{array}\right)\right\}\)
5.
\(S=\left\{ \left(\begin{array}{r} 5\\2\\-1\\4 \end{array}\right), \left(\begin{array}{r} 0 \\ 2 \\ 3 \\ 1\end{array}\right)\right\}\)

Exercise Group.

Determine if the given vector \(\vec{v}\) is in the span of the vectors in the set \(S\text{.}\) Do so without doing elimination if possible.
6.
\(\vec{v}=\left(\begin{array}{r} -3\\5 \end{array}\right), S=\left\{ \left(\begin{array}{r} 1\\-1 \end{array}\right), \left(\begin{array}{r} 1 \\ 0 \end{array}\right)\right\}\)
7.
\(\vec{v}=\left(\begin{array}{r} -1\\6\\8 \end{array}\right), S=\left\{ \left(\begin{array}{r} 2\\3\\-1 \end{array}\right), \left(\begin{array}{r} 1 \\ 0 \\ -2 \end{array}\right)\right\}\)
8.
\(\vec{v}=\left(\begin{array}{r} 3\\-1\\4 \end{array}\right), S=\left\{ \left(\begin{array}{r} 1\\0\\-1 \end{array}\right), \left(\begin{array}{r} -1 \\ 0 \\ 1 \end{array}\right), \left(\begin{array}{r} 0 \\ 0 \\ -1 \end{array}\right)\right\}\)
9.
\(\vec{v}=\left(\begin{array}{r} 0\\2\\-1\\5 \end{array}\right), S=\left\{ \left(\begin{array}{r} -1\\2\\-1\\0 \end{array}\right), \left(\begin{array}{r} 1 \\ 3 \\ 2\\-1 \end{array}\right)\right\}\)

Exercise Group.

Determine if the set of vectors \(S\) is a basis of the given vector space \(V\text{.}\) Explain.
10.
\(S=\left\{ \left(\begin{array}{r} 1\\-3 \end{array}\right), \left(\begin{array}{r} 2\\-1 \end{array}\right) \right\}, V=\R^2\)
11.
\(S=\left\{ \left(\begin{array}{r} 7\\-1\\2 \end{array}\right), \left(\begin{array}{r} 0\\4\\-1 \end{array}\right) \right\}, V=\R^3\)
12.
\(S=\left\{ \left(\begin{array}{r} -1\\2\\4 \end{array}\right), \left(\begin{array}{r} 3\\6\\2 \end{array}\right), \left(\begin{array}{r} 7\\22\\14 \end{array}\right) \right\}, V=\R^3\)
13.
\(S=\left\{ \left(\begin{array}{r} 1\\-3\\2 \end{array}\right), \left(\begin{array}{r} 2\\0\\1 \end{array}\right), \left(\begin{array}{r} 3\\-2\\4 \end{array}\right) \right\}, V=\R^3\)

14.

Find a basis for \(\R^3\) other than \(\left\{\left. \left(\begin{array}{r} \pm k\\ 0 \\0 \end{array}\right), \left(\begin{array}{r} 0\\ \pm k \\0 \end{array}\right), \left(\begin{array}{r} 0\\ 0 \\\pm k \end{array}\right)\,\right|k\ne0 \right\}\) (or any other set already shown to be a basis in a previous exercise). Show your set is a basis.

15.

Give a set of three vectors in \(\R^3\) that are NOT a basis for \(\R^3\text{.}\) Explain how you know.

16.

How many vectors will be in a basis for \(\R^5\text{?}\) Explain.

18.

Prove or disprove: Let \(V\) be a vector space and \(\left\{\vec{q_1},\vec{q_2},\ldots,\vec{q_n}\right\}\) be an orthonormal basis for \(V.\) Then any subspace of \(V\) has a basis that is a subset of \(\left\{\vec{q_1},\vec{q_2},\ldots,\vec{q_n}\right\}.\)
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