22M:100:001 (MATH:3600:0001) Quiz 2 April 3, 2013

1.) Determine which of the following sets of vectors are linearly dependent versus linearly independent. Circle the correct answer

[10] 1i.) \{( \begin{matrix}1 \\ 2 \end{matrix} ) , ( \begin{matrix}−1 \\ 3 \end{matrix} ) , ( \begin{matrix}4 \\ 5 \end{matrix} )\}

a.) linearly dependent b.) linearly independent

[10] 1ii.) \{( \begin{matrix}1 \\ 2 \\ 3 \end{matrix} ) , ( \begin{matrix}−1 \\ −2 \\ −3 \end{matrix} )\}

a.) linearly dependent b.) linearly independent

[10] 1iii.) \{( \begin{matrix}1 \\ 2 \\ 4 \end{matrix} ) , ( \begin{matrix}−1 \\ 3 \\ 4 \end{matrix} )\}

a.) linearly dependent b.) linearly independent

[10] 1iv.) \{( \begin{matrix}1 \\ 2 \\ 4 \end{matrix} ) , ( \begin{matrix}1 \\ 3 \\ 2 \end{matrix} ) , ( \begin{matrix}2 \\ 5 \\ 6 \end{matrix} )\}

a.) linearly dependent b.) linearly independent

[10] 1v.) \{( \begin{matrix}1 \\ 2 \\ 0 \end{matrix} ) , ( \begin{matrix}1 \\ 3 \\ 0 \end{matrix} ) , ( \begin{matrix}2 \\ 3 \\ 6 \end{matrix} )\}

a.) linearly dependent b.) linearly independent

2.) If A ( \begin{matrix}1 \\ 2 \end{matrix} ) = ( \begin{matrix}5 \\ 5 \end{matrix} ), A ( \begin{matrix}2 \\ 1 \end{matrix} ) = ( \begin{matrix}4 \\ 4 \end{matrix} ), A ( \begin{matrix}2 \\ 2 \end{matrix} ) = ( \begin{matrix}6 \\ 6 \end{matrix} ), A ( \begin{matrix}2 \\ 3 \end{matrix} ) = ( \begin{matrix}8 \\ 8 \end{matrix} ), A ( \begin{matrix}3 \\ 2 \end{matrix} ) = ( \begin{matrix}7 \\ 7 \end{matrix} ).

[10] 2a.) An eigenvalue of A is

[15] 2b.) 4 eigenvectors corresponding to this eigenvalue are

[15] 3a.) Find the eigenvalues of A = \{( \begin{matrix}1 & 1 \\ 2 & −1 \end{matrix} )

[10] 3b.) Find one eigenvector corresponding to each eigenvalue.

An eigenvalue of A is and an eigenvector corresponding to this eigenvalue is

An eigenvalue of A is and an eigenvector corresponding to this eigenvalue is .

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