CoCalc -- worksheet8.tex

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\documentclass{exam}
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\usepackage{hyperref}
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\usepackage{amsmath}
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\usepackage{amsfonts}
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\begin{document}
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\begin{center}
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    Worksheet 8 - Never due
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\end{center}
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\begin{questions}
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    \question
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    Give an example of each of the following. If it is not possible, write
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    ``NOT POSSIBLE''.
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    \begin{parts}
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        \part
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        Give an example of a basis of $\mathbb{R}^4$ such that each element
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        lies in the hyperplane $2w+3x+y+z=0$.
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        \part
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        Give an example of a basis of $\mathbb{R}^4$ such that each element
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        lies in the hyperplane $2w+3x+y+z=1$.
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        \part
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        Give an example of a matrix that is orthogonally diagonalizable but not
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        diagonalizable.
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        \part
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        Give an example of a matrix that is diagonalizable but not orthogonally
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        diagonalizable.
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        \part
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        Give an example of a nonzero matrix $A$ such that $A^2=0$.
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        \part
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        Give an example of a nonzero matrix $A$ such that $A^2=I$.
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        \part
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        Give an example of a nonzero matrix $A$ such that $A^2=I$ and the
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        nullity of $A$ is 1.
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        \part
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        Give an example of an orthogonal set that is not linearly independent.
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        \part
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        Give an example of an orthogonal set that is not spanning.
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        \part
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        Give an example of a $2\times 3$ matrix whose rank is equal to its
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        nullity.
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        \part
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        Give an example of 2 matrices $A$ and $B$ such that $A^3=B^3$.
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        \part
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        Give an example of 2 matrices $A$ and $B$ such that $A$ and $B$ each have
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        nullity 1 but $AB$ has nullity 0.
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        \part
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        Give an example of 2 matrices $A$ and $B$ such that $A$ and $B$ each have
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        nullity 0 but $AB$ has nullity 1.
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        \part
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        Give an example of a diagonalizable matrix that is not invertible.
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        \part
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        Give an example of an invertible matrix that is not diagonalizable.
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        \part
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        Give an example of a symmetric matrix that is not diagonalizable.
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        \part
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        Give an example of a symmetric matrix that is not invertible.
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        \part
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        Give an example of an orthogonal matrix that is not invertible.
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        \part
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        Give an example of an invertible matrix that is not orthogonal.
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        \part
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        Give an example of a matrix with distinct eigenvalues that is not
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        invertible.
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        \part
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        Give an example of a $3\times 3$ orthogonal matrix with only one eigenvalue.
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        \part
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        Give an example of a $3\times 3$ matrix whose only eigenvalue is 2.
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        \part
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        Give an example of a $3\times 3$ invertible matrix whose only
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        eigenvalue is 2.
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        \part
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        Give an example of a matrix $A$ and an eigenvalue $\lambda$ such that
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        the algebraic multiplicity of $\lambda$ is less than the geometric
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        multiplicity.
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        \part
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        Give an example of a matrix $A$ and an eigenvalue $\lambda$ such that
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        the geometric multiplicity of $\lambda$ is less than the algebraic
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        multiplicity.
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        \part
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        Give an example of a matrix $A$ and an eigenvalue $\lambda$ such that
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        the eigenspace is 0-dimensional.
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   \end{parts}
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\end{questions}
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\end{document}
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