Explain how to find a basis for the eigenspace associated to the eigenvalue \( -4 \) in the matrix
\[ \left[\begin{array}{cccc} -6 & 8 & -6 & -4 \\ 1 & -8 & 3 & 2 \\ 1 & -4 & -1 & 2 \\ 1 & -4 & 3 & -2 \end{array}\right] \]
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} -2 & 8 & -6 & -4 \\ 1 & -4 & 3 & 2 \\ 1 & -4 & 3 & 2 \\ 1 & -4 & 3 & 2 \end{array}\right] = \left[\begin{array}{cccc} 1 & -4 & 3 & 2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
A basis of the eigenspace is \( \left\{ \left[\begin{array}{c} 4 \\ 1 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} -3 \\ 0 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -2 \\ 0 \\ 0 \\ 1 \end{array}\right] \right\} \).
Explain how to find a basis for the eigenspace associated to the eigenvalue \( -2 \) in the matrix
\[ \left[\begin{array}{cccc} -1 & -1 & -4 & -4 \\ -1 & -4 & 1 & -2 \\ 0 & 4 & 2 & 8 \\ 0 & -2 & -2 & -6 \end{array}\right] \]
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 1 & -1 & -4 & -4 \\ -1 & -2 & 1 & -2 \\ 0 & 4 & 4 & 8 \\ 0 & -2 & -2 & -4 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -3 & -2 \\ 0 & 1 & 1 & 2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
A basis of the eigenspace is \( \left\{ \left[\begin{array}{c} 3 \\ -1 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} 2 \\ -2 \\ 0 \\ 1 \end{array}\right] \right\} \).
Explain how to find a basis for the eigenspace associated to the eigenvalue \( 2 \) in the matrix
\[ \left[\begin{array}{cccc} 5 & -6 & 3 & -3 \\ 2 & -2 & 2 & -2 \\ -1 & 2 & 1 & 1 \\ -4 & 8 & -4 & 6 \end{array}\right] \]
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 3 & -6 & 3 & -3 \\ 2 & -4 & 2 & -2 \\ -1 & 2 & -1 & 1 \\ -4 & 8 & -4 & 4 \end{array}\right] = \left[\begin{array}{cccc} 1 & -2 & 1 & -1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
A basis of the eigenspace is \( \left\{ \left[\begin{array}{c} 2 \\ 1 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} -1 \\ 0 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} 1 \\ 0 \\ 0 \\ 1 \end{array}\right] \right\} \).
Explain how to find a basis for the eigenspace associated to the eigenvalue \( 1 \) in the matrix
\[ \left[\begin{array}{cccc} 3 & 3 & 2 & -7 \\ -1 & 0 & -1 & 3 \\ -3 & -2 & -1 & 5 \\ 2 & 8 & 0 & -5 \end{array}\right] \]
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 2 & 3 & 2 & -7 \\ -1 & -1 & -1 & 3 \\ -3 & -2 & -2 & 5 \\ 2 & 8 & 0 & -6 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & -3 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
A basis of the eigenspace is \( \left\{ \left[\begin{array}{c} -1 \\ 1 \\ 3 \\ 1 \end{array}\right] \right\} \).
Explain how to find a basis for the eigenspace associated to the eigenvalue \( -3 \) in the matrix
\[ \left[\begin{array}{cccc} -6 & 3 & 0 & 6 \\ -1 & -2 & 0 & 2 \\ 2 & -2 & -3 & -4 \\ -4 & 4 & 0 & 5 \end{array}\right] \]
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} -3 & 3 & 0 & 6 \\ -1 & 1 & 0 & 2 \\ 2 & -2 & 0 & -4 \\ -4 & 4 & 0 & 8 \end{array}\right] = \left[\begin{array}{cccc} 1 & -1 & 0 & -2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
A basis of the eigenspace is \( \left\{ \left[\begin{array}{c} 1 \\ 1 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} 0 \\ 0 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} 2 \\ 0 \\ 0 \\ 1 \end{array}\right] \right\} \).
Explain how to find a basis for the eigenspace associated to the eigenvalue \( 2 \) in the matrix
\[ \left[\begin{array}{cccc} 3 & 1 & 0 & 1 \\ 0 & 3 & -1 & 5 \\ 1 & 0 & 4 & -7 \\ 3 & 2 & 1 & 0 \end{array}\right] \]
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 1 & 1 & 0 & 1 \\ 0 & 1 & -1 & 5 \\ 1 & 0 & 2 & -7 \\ 3 & 2 & 1 & -2 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & -1 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & -3 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
A basis of the eigenspace is \( \left\{ \left[\begin{array}{c} 1 \\ -2 \\ 3 \\ 1 \end{array}\right] \right\} \).
Explain how to find a basis for the eigenspace associated to the eigenvalue \( 3 \) in the matrix
\[ \left[\begin{array}{cccc} 4 & -1 & -1 & -1 \\ 1 & 3 & 0 & 1 \\ 5 & -2 & 1 & 1 \\ 2 & -3 & -3 & -1 \end{array}\right] \]
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 1 & -1 & -1 & -1 \\ 1 & 0 & 0 & 1 \\ 5 & -2 & -2 & 1 \\ 2 & -3 & -3 & -4 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
A basis of the eigenspace is \( \left\{ \left[\begin{array}{c} 0 \\ -1 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -1 \\ -2 \\ 0 \\ 1 \end{array}\right] \right\} \).
Explain how to find a basis for the eigenspace associated to the eigenvalue \( 1 \) in the matrix
\[ \left[\begin{array}{cccc} 2 & -2 & -5 & 6 \\ 1 & -1 & -5 & 6 \\ -1 & 3 & 7 & -8 \\ 0 & -1 & -1 & 3 \end{array}\right] \]
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 1 & -2 & -5 & 6 \\ 1 & -2 & -5 & 6 \\ -1 & 3 & 6 & -8 \\ 0 & -1 & -1 & 2 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -3 & 2 \\ 0 & 1 & 1 & -2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
A basis of the eigenspace is \( \left\{ \left[\begin{array}{c} 3 \\ -1 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -2 \\ 2 \\ 0 \\ 1 \end{array}\right] \right\} \).
Explain how to find a basis for the eigenspace associated to the eigenvalue \( 4 \) in the matrix
\[ \left[\begin{array}{cccc} 3 & -1 & 0 & -1 \\ 1 & 4 & 1 & 1 \\ -2 & 0 & 2 & -2 \\ -1 & 1 & -2 & 3 \end{array}\right] \]
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} -1 & -1 & 0 & -1 \\ 1 & 0 & 1 & 1 \\ -2 & 0 & -2 & -2 \\ -1 & 1 & -2 & -1 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 1 & 1 \\ 0 & 1 & -1 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
A basis of the eigenspace is \( \left\{ \left[\begin{array}{c} -1 \\ 1 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -1 \\ 0 \\ 0 \\ 1 \end{array}\right] \right\} \).
Explain how to find a basis for the eigenspace associated to the eigenvalue \( -1 \) in the matrix
\[ \left[\begin{array}{cccc} -1 & 1 & 3 & 2 \\ -1 & 0 & -1 & 1 \\ 3 & -3 & 3 & -2 \\ 3 & -1 & 5 & -4 \end{array}\right] \]
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 0 & 1 & 3 & 2 \\ -1 & 1 & -1 & 1 \\ 3 & -3 & 4 & -2 \\ 3 & -1 & 5 & -3 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & -3 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
A basis of the eigenspace is \( \left\{ \left[\begin{array}{c} 3 \\ 1 \\ -1 \\ 1 \end{array}\right] \right\} \).
Explain how to find a basis for the eigenspace associated to the eigenvalue \( 4 \) in the matrix
\[ \left[\begin{array}{cccc} 2 & 1 & 0 & -4 \\ 5 & 0 & 5 & 8 \\ 3 & -2 & 6 & 5 \\ 0 & 1 & -7 & 9 \end{array}\right] \]
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} -2 & 1 & 0 & -4 \\ 5 & -4 & 5 & 8 \\ 3 & -2 & 2 & 5 \\ 0 & 1 & -7 & 5 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & -2 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
A basis of the eigenspace is \( \left\{ \left[\begin{array}{c} -1 \\ 2 \\ 1 \\ 1 \end{array}\right] \right\} \).
Explain how to find a basis for the eigenspace associated to the eigenvalue \( -1 \) in the matrix
\[ \left[\begin{array}{cccc} -1 & -2 & -4 & -2 \\ 1 & 2 & 4 & 5 \\ -1 & 2 & 5 & 0 \\ -1 & -4 & -6 & -7 \end{array}\right] \]
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 0 & -2 & -4 & -2 \\ 1 & 3 & 4 & 5 \\ -1 & 2 & 6 & 0 \\ -1 & -4 & -6 & -6 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -2 & 2 \\ 0 & 1 & 2 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
A basis of the eigenspace is \( \left\{ \left[\begin{array}{c} 2 \\ -2 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -2 \\ -1 \\ 0 \\ 1 \end{array}\right] \right\} \).
Explain how to find a basis for the eigenspace associated to the eigenvalue \( 2 \) in the matrix
\[ \left[\begin{array}{cccc} 1 & 3 & -2 & 3 \\ 1 & 0 & 1 & -3 \\ -1 & 6 & -3 & 3 \\ -2 & 6 & -4 & 8 \end{array}\right] \]
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} -1 & 3 & -2 & 3 \\ 1 & -2 & 1 & -3 \\ -1 & 6 & -5 & 3 \\ -2 & 6 & -4 & 6 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -1 & -3 \\ 0 & 1 & -1 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
A basis of the eigenspace is \( \left\{ \left[\begin{array}{c} 1 \\ 1 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} 3 \\ 0 \\ 0 \\ 1 \end{array}\right] \right\} \).
Explain how to find a basis for the eigenspace associated to the eigenvalue \( 3 \) in the matrix
\[ \left[\begin{array}{cccc} 4 & -4 & 6 & -8 \\ 0 & 3 & 1 & -1 \\ -2 & 8 & 6 & 1 \\ 0 & 0 & -5 & 8 \end{array}\right] \]
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 1 & -4 & 6 & -8 \\ 0 & 0 & 1 & -1 \\ -2 & 8 & 3 & 1 \\ 0 & 0 & -5 & 5 \end{array}\right] = \left[\begin{array}{cccc} 1 & -4 & 0 & -2 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
A basis of the eigenspace is \( \left\{ \left[\begin{array}{c} 4 \\ 1 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} 2 \\ 0 \\ 1 \\ 1 \end{array}\right] \right\} \).
Explain how to find a basis for the eigenspace associated to the eigenvalue \( 4 \) in the matrix
\[ \left[\begin{array}{cccc} 2 & 0 & 0 & -2 \\ 3 & 4 & 0 & 3 \\ 5 & 0 & 4 & 5 \\ -5 & 0 & 0 & -1 \end{array}\right] \]
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} -2 & 0 & 0 & -2 \\ 3 & 0 & 0 & 3 \\ 5 & 0 & 0 & 5 \\ -5 & 0 & 0 & -5 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
A basis of the eigenspace is \( \left\{ \left[\begin{array}{c} 0 \\ 1 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} 0 \\ 0 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -1 \\ 0 \\ 0 \\ 1 \end{array}\right] \right\} \).
Explain how to find a basis for the eigenspace associated to the eigenvalue \( -3 \) in the matrix
\[ \left[\begin{array}{cccc} -4 & -3 & 5 & 7 \\ 1 & -1 & -4 & -5 \\ 0 & -3 & 0 & 6 \\ -2 & -2 & 6 & 3 \end{array}\right] \]
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} -1 & -3 & 5 & 7 \\ 1 & 2 & -4 & -5 \\ 0 & -3 & 3 & 6 \\ -2 & -2 & 6 & 6 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -2 & -1 \\ 0 & 1 & -1 & -2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
A basis of the eigenspace is \( \left\{ \left[\begin{array}{c} 2 \\ 1 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} 1 \\ 2 \\ 0 \\ 1 \end{array}\right] \right\} \).
Explain how to find a basis for the eigenspace associated to the eigenvalue \( -3 \) in the matrix
\[ \left[\begin{array}{cccc} 1 & 1 & 2 & 6 \\ 3 & -2 & 2 & 5 \\ 0 & 3 & 3 & 6 \\ 5 & -4 & -8 & -6 \end{array}\right] \]
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 4 & 1 & 2 & 6 \\ 3 & 1 & 2 & 5 \\ 0 & 3 & 6 & 6 \\ 5 & -4 & -8 & -3 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 1 \\ 0 & 1 & 2 & 2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
A basis of the eigenspace is \( \left\{ \left[\begin{array}{c} 0 \\ -2 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -1 \\ -2 \\ 0 \\ 1 \end{array}\right] \right\} \).
Explain how to find a basis for the eigenspace associated to the eigenvalue \( -3 \) in the matrix
\[ \left[\begin{array}{cccc} -2 & 2 & 2 & 2 \\ -1 & -1 & 6 & 6 \\ -3 & 1 & 5 & 8 \\ 0 & 1 & 2 & -1 \end{array}\right] \]
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 1 & 2 & 2 & 2 \\ -1 & 2 & 6 & 6 \\ -3 & 1 & 8 & 8 \\ 0 & 1 & 2 & 2 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -2 & -2 \\ 0 & 1 & 2 & 2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
A basis of the eigenspace is \( \left\{ \left[\begin{array}{c} 2 \\ -2 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} 2 \\ -2 \\ 0 \\ 1 \end{array}\right] \right\} \).
Explain how to find a basis for the eigenspace associated to the eigenvalue \( 1 \) in the matrix
\[ \left[\begin{array}{cccc} 1 & 0 & 3 & 0 \\ -1 & 4 & -1 & 3 \\ -1 & 3 & 2 & 3 \\ 1 & -3 & 2 & -2 \end{array}\right] \]
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 0 & 0 & 3 & 0 \\ -1 & 3 & -1 & 3 \\ -1 & 3 & 1 & 3 \\ 1 & -3 & 2 & -3 \end{array}\right] = \left[\begin{array}{cccc} 1 & -3 & 0 & -3 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
A basis of the eigenspace is \( \left\{ \left[\begin{array}{c} 3 \\ 1 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} 3 \\ 0 \\ 0 \\ 1 \end{array}\right] \right\} \).
Explain how to find a basis for the eigenspace associated to the eigenvalue \( -1 \) in the matrix
\[ \left[\begin{array}{cccc} 6 & 7 & 0 & 7 \\ -4 & -5 & 0 & -4 \\ 3 & 3 & -1 & 3 \\ 4 & 4 & 0 & 3 \end{array}\right] \]
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 7 & 7 & 0 & 7 \\ -4 & -4 & 0 & -4 \\ 3 & 3 & 0 & 3 \\ 4 & 4 & 0 & 4 \end{array}\right] = \left[\begin{array}{cccc} 1 & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
A basis of the eigenspace is \( \left\{ \left[\begin{array}{c} -1 \\ 1 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} 0 \\ 0 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -1 \\ 0 \\ 0 \\ 1 \end{array}\right] \right\} \).
Explain how to find a basis for the eigenspace associated to the eigenvalue \( -3 \) in the matrix
\[ \left[\begin{array}{cccc} -2 & 4 & -5 & -1 \\ 1 & 1 & -4 & -1 \\ -1 & -4 & -2 & 1 \\ -2 & -8 & 5 & -1 \end{array}\right] \]
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 1 & 4 & -5 & -1 \\ 1 & 4 & -4 & -1 \\ -1 & -4 & 1 & 1 \\ -2 & -8 & 5 & 2 \end{array}\right] = \left[\begin{array}{cccc} 1 & 4 & 0 & -1 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
A basis of the eigenspace is \( \left\{ \left[\begin{array}{c} -4 \\ 1 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} 1 \\ 0 \\ 0 \\ 1 \end{array}\right] \right\} \).
Explain how to find a basis for the eigenspace associated to the eigenvalue \( 2 \) in the matrix
\[ \left[\begin{array}{cccc} 1 & -3 & 1 & 3 \\ -1 & 0 & 1 & 2 \\ -2 & -4 & 5 & 1 \\ -2 & -6 & 3 & 5 \end{array}\right] \]
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} -1 & -3 & 1 & 3 \\ -1 & -2 & 1 & 2 \\ -2 & -4 & 3 & 1 \\ -2 & -6 & 3 & 3 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & -3 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & -3 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
A basis of the eigenspace is \( \left\{ \left[\begin{array}{c} 3 \\ 1 \\ 3 \\ 1 \end{array}\right] \right\} \).
Explain how to find a basis for the eigenspace associated to the eigenvalue \( -1 \) in the matrix
\[ \left[\begin{array}{cccc} 1 & 1 & 2 & 2 \\ -1 & -1 & -1 & -2 \\ 0 & 2 & -1 & -4 \\ -4 & -5 & -4 & 1 \end{array}\right] \]
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 2 & 1 & 2 & 2 \\ -1 & 0 & -1 & -2 \\ 0 & 2 & 0 & -4 \\ -4 & -5 & -4 & 2 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 1 & 2 \\ 0 & 1 & 0 & -2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
A basis of the eigenspace is \( \left\{ \left[\begin{array}{c} -1 \\ 0 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -2 \\ 2 \\ 0 \\ 1 \end{array}\right] \right\} \).
Explain how to find a basis for the eigenspace associated to the eigenvalue \( -3 \) in the matrix
\[ \left[\begin{array}{cccc} 1 & 1 & 3 & 6 \\ -2 & -4 & -1 & -4 \\ -3 & 1 & -7 & -1 \\ 1 & 0 & 1 & -2 \end{array}\right] \]
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 4 & 1 & 3 & 6 \\ -2 & -1 & -1 & -4 \\ -3 & 1 & -4 & -1 \\ 1 & 0 & 1 & 1 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 1 & 1 \\ 0 & 1 & -1 & 2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
A basis of the eigenspace is \( \left\{ \left[\begin{array}{c} -1 \\ 1 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -1 \\ -2 \\ 0 \\ 1 \end{array}\right] \right\} \).
Explain how to find a basis for the eigenspace associated to the eigenvalue \( 1 \) in the matrix
\[ \left[\begin{array}{cccc} 0 & -1 & 3 & 0 \\ 1 & 1 & -1 & -2 \\ 0 & 0 & 1 & 0 \\ -1 & -1 & 3 & 1 \end{array}\right] \]
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} -1 & -1 & 3 & 0 \\ 1 & 0 & -1 & -2 \\ 0 & 0 & 0 & 0 \\ -1 & -1 & 3 & 0 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -1 & -2 \\ 0 & 1 & -2 & 2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
A basis of the eigenspace is \( \left\{ \left[\begin{array}{c} 1 \\ 2 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} 2 \\ -2 \\ 0 \\ 1 \end{array}\right] \right\} \).
Explain how to find a basis for the eigenspace associated to the eigenvalue \( 1 \) in the matrix
\[ \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ -2 & -7 & 1 & -3 \\ -1 & -4 & 1 & -3 \\ 2 & 8 & -1 & 4 \end{array}\right] \]
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 0 & 0 & 0 & 0 \\ -2 & -8 & 1 & -3 \\ -1 & -4 & 0 & -3 \\ 2 & 8 & -1 & 3 \end{array}\right] = \left[\begin{array}{cccc} 1 & 4 & 0 & 3 \\ 0 & 0 & 1 & 3 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
A basis of the eigenspace is \( \left\{ \left[\begin{array}{c} -4 \\ 1 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} -3 \\ 0 \\ -3 \\ 1 \end{array}\right] \right\} \).
Explain how to find a basis for the eigenspace associated to the eigenvalue \( 2 \) in the matrix
\[ \left[\begin{array}{cccc} -2 & 7 & -3 & 2 \\ 3 & 0 & -1 & 5 \\ 5 & -8 & 5 & -1 \\ 0 & 4 & -4 & 10 \end{array}\right] \]
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} -4 & 7 & -3 & 2 \\ 3 & -2 & -1 & 5 \\ 5 & -8 & 3 & -1 \\ 0 & 4 & -4 & 8 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -1 & 3 \\ 0 & 1 & -1 & 2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
A basis of the eigenspace is \( \left\{ \left[\begin{array}{c} 1 \\ 1 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -3 \\ -2 \\ 0 \\ 1 \end{array}\right] \right\} \).
Explain how to find a basis for the eigenspace associated to the eigenvalue \( 4 \) in the matrix
\[ \left[\begin{array}{cccc} 6 & 6 & 4 & 4 \\ -1 & 1 & -2 & -2 \\ 0 & 0 & 4 & 0 \\ -1 & -3 & -2 & 2 \end{array}\right] \]
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 2 & 6 & 4 & 4 \\ -1 & -3 & -2 & -2 \\ 0 & 0 & 0 & 0 \\ -1 & -3 & -2 & -2 \end{array}\right] = \left[\begin{array}{cccc} 1 & 3 & 2 & 2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
A basis of the eigenspace is \( \left\{ \left[\begin{array}{c} -3 \\ 1 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} -2 \\ 0 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -2 \\ 0 \\ 0 \\ 1 \end{array}\right] \right\} \).
Explain how to find a basis for the eigenspace associated to the eigenvalue \( 1 \) in the matrix
\[ \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 1 & -1 & -2 & -3 \\ 0 & 0 & 1 & 0 \\ 2 & -4 & -4 & -5 \end{array}\right] \]
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 0 & 0 & 0 & 0 \\ 1 & -2 & -2 & -3 \\ 0 & 0 & 0 & 0 \\ 2 & -4 & -4 & -6 \end{array}\right] = \left[\begin{array}{cccc} 1 & -2 & -2 & -3 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
A basis of the eigenspace is \( \left\{ \left[\begin{array}{c} 2 \\ 1 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} 2 \\ 0 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} 3 \\ 0 \\ 0 \\ 1 \end{array}\right] \right\} \).
Explain how to find a basis for the eigenspace associated to the eigenvalue \( 4 \) in the matrix
\[ \left[\begin{array}{cccc} 5 & -3 & 5 & 8 \\ 0 & 5 & -2 & -3 \\ 1 & -2 & 7 & 5 \\ 0 & 1 & -2 & 1 \end{array}\right] \]
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 1 & -3 & 5 & 8 \\ 0 & 1 & -2 & -3 \\ 1 & -2 & 3 & 5 \\ 0 & 1 & -2 & -3 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -1 & -1 \\ 0 & 1 & -2 & -3 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
A basis of the eigenspace is \( \left\{ \left[\begin{array}{c} 1 \\ 2 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} 1 \\ 3 \\ 0 \\ 1 \end{array}\right] \right\} \).
Explain how to find a basis for the eigenspace associated to the eigenvalue \( 1 \) in the matrix
\[ \left[\begin{array}{cccc} 2 & 1 & -1 & 1 \\ -4 & -2 & 4 & -3 \\ 3 & -2 & -2 & -2 \\ 5 & 0 & -5 & 1 \end{array}\right] \]
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 1 & 1 & -1 & 1 \\ -4 & -3 & 4 & -3 \\ 3 & -2 & -3 & -2 \\ 5 & 0 & -5 & 0 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -1 & 0 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
A basis of the eigenspace is \( \left\{ \left[\begin{array}{c} 1 \\ 0 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} 0 \\ -1 \\ 0 \\ 1 \end{array}\right] \right\} \).
Explain how to find a basis for the eigenspace associated to the eigenvalue \( -4 \) in the matrix
\[ \left[\begin{array}{cccc} -3 & 1 & 2 & 5 \\ 0 & -3 & 4 & 6 \\ 0 & 1 & 1 & 8 \\ 0 & -1 & -3 & -8 \end{array}\right] \]
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 1 & 1 & 2 & 5 \\ 0 & 1 & 4 & 6 \\ 0 & 1 & 5 & 8 \\ 0 & -1 & -3 & -4 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 3 \\ 0 & 1 & 0 & -2 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
A basis of the eigenspace is \( \left\{ \left[\begin{array}{c} -3 \\ 2 \\ -2 \\ 1 \end{array}\right] \right\} \).
Explain how to find a basis for the eigenspace associated to the eigenvalue \( -1 \) in the matrix
\[ \left[\begin{array}{cccc} -2 & 2 & 0 & 1 \\ 0 & 0 & 1 & 1 \\ 1 & -3 & -2 & -2 \\ 0 & -1 & -1 & -2 \end{array}\right] \]
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} -1 & 2 & 0 & 1 \\ 0 & 1 & 1 & 1 \\ 1 & -3 & -1 & -2 \\ 0 & -1 & -1 & -1 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 2 & 1 \\ 0 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
A basis of the eigenspace is \( \left\{ \left[\begin{array}{c} -2 \\ -1 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -1 \\ -1 \\ 0 \\ 1 \end{array}\right] \right\} \).
Explain how to find a basis for the eigenspace associated to the eigenvalue \( -2 \) in the matrix
\[ \left[\begin{array}{cccc} -1 & -1 & 5 & 4 \\ -1 & -1 & -6 & -5 \\ 0 & 1 & -2 & 1 \\ 1 & 0 & 0 & -2 \end{array}\right] \]
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 1 & -1 & 5 & 4 \\ -1 & 1 & -6 & -5 \\ 0 & 1 & 0 & 1 \\ 1 & 0 & 0 & 0 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
A basis of the eigenspace is \( \left\{ \left[\begin{array}{c} 0 \\ -1 \\ -1 \\ 1 \end{array}\right] \right\} \).
Explain how to find a basis for the eigenspace associated to the eigenvalue \( -2 \) in the matrix
\[ \left[\begin{array}{cccc} -4 & -3 & -6 & -4 \\ -1 & -4 & -4 & -3 \\ 0 & 0 & -2 & 0 \\ -2 & -3 & -6 & -6 \end{array}\right] \]
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} -2 & -3 & -6 & -4 \\ -1 & -2 & -4 & -3 \\ 0 & 0 & 0 & 0 \\ -2 & -3 & -6 & -4 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & -1 \\ 0 & 1 & 2 & 2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
A basis of the eigenspace is \( \left\{ \left[\begin{array}{c} 0 \\ -2 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} 1 \\ -2 \\ 0 \\ 1 \end{array}\right] \right\} \).
Explain how to find a basis for the eigenspace associated to the eigenvalue \( -2 \) in the matrix
\[ \left[\begin{array}{cccc} -1 & 4 & 2 & 3 \\ 0 & -1 & 0 & 2 \\ 0 & -3 & -1 & -7 \\ -2 & -5 & -2 & -4 \end{array}\right] \]
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 1 & 4 & 2 & 3 \\ 0 & 1 & 0 & 2 \\ 0 & -3 & 1 & -7 \\ -2 & -5 & -2 & -2 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & -3 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
A basis of the eigenspace is \( \left\{ \left[\begin{array}{c} 3 \\ -2 \\ 1 \\ 1 \end{array}\right] \right\} \).
Explain how to find a basis for the eigenspace associated to the eigenvalue \( 3 \) in the matrix
\[ \left[\begin{array}{cccc} 2 & -2 & -3 & -2 \\ 2 & 7 & 6 & 4 \\ 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 3 \end{array}\right] \]
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} -1 & -2 & -3 & -2 \\ 2 & 4 & 6 & 4 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] = \left[\begin{array}{cccc} 1 & 2 & 3 & 2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
A basis of the eigenspace is \( \left\{ \left[\begin{array}{c} -2 \\ 1 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} -3 \\ 0 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -2 \\ 0 \\ 0 \\ 1 \end{array}\right] \right\} \).
Explain how to find a basis for the eigenspace associated to the eigenvalue \( 4 \) in the matrix
\[ \left[\begin{array}{cccc} 5 & -3 & 7 & 7 \\ 0 & 5 & -3 & -4 \\ 0 & 2 & -2 & -7 \\ 0 & 1 & -3 & 0 \end{array}\right] \]
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 1 & -3 & 7 & 7 \\ 0 & 1 & -3 & -4 \\ 0 & 2 & -6 & -7 \\ 0 & 1 & -3 & -4 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -2 & 0 \\ 0 & 1 & -3 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
A basis of the eigenspace is \( \left\{ \left[\begin{array}{c} 2 \\ 3 \\ 1 \\ 0 \end{array}\right] \right\} \).
Explain how to find a basis for the eigenspace associated to the eigenvalue \( -3 \) in the matrix
\[ \left[\begin{array}{cccc} -4 & 5 & -1 & 2 \\ 1 & -8 & 0 & -1 \\ -1 & 5 & -5 & 3 \\ 1 & -5 & -6 & 2 \end{array}\right] \]
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} -1 & 5 & -1 & 2 \\ 1 & -5 & 0 & -1 \\ -1 & 5 & -2 & 3 \\ 1 & -5 & -6 & 5 \end{array}\right] = \left[\begin{array}{cccc} 1 & -5 & 0 & -1 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
A basis of the eigenspace is \( \left\{ \left[\begin{array}{c} 5 \\ 1 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} 1 \\ 0 \\ 1 \\ 1 \end{array}\right] \right\} \).
Explain how to find a basis for the eigenspace associated to the eigenvalue \( -3 \) in the matrix
\[ \left[\begin{array}{cccc} -2 & -1 & 2 & 4 \\ -2 & -1 & -4 & -8 \\ 1 & -1 & -1 & 4 \\ 1 & -1 & 2 & 1 \end{array}\right] \]
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 1 & -1 & 2 & 4 \\ -2 & 2 & -4 & -8 \\ 1 & -1 & 2 & 4 \\ 1 & -1 & 2 & 4 \end{array}\right] = \left[\begin{array}{cccc} 1 & -1 & 2 & 4 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
A basis of the eigenspace is \( \left\{ \left[\begin{array}{c} 1 \\ 1 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} -2 \\ 0 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -4 \\ 0 \\ 0 \\ 1 \end{array}\right] \right\} \).
Explain how to find a basis for the eigenspace associated to the eigenvalue \( 4 \) in the matrix
\[ \left[\begin{array}{cccc} 5 & -1 & 0 & -1 \\ 4 & 1 & 3 & -8 \\ 1 & 0 & 8 & -6 \\ -1 & 1 & 3 & 2 \end{array}\right] \]
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 1 & -1 & 0 & -1 \\ 4 & -3 & 3 & -8 \\ 1 & 0 & 4 & -6 \\ -1 & 1 & 3 & -2 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & -2 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
A basis of the eigenspace is \( \left\{ \left[\begin{array}{c} 2 \\ 1 \\ 1 \\ 1 \end{array}\right] \right\} \).
Explain how to find a basis for the eigenspace associated to the eigenvalue \( 3 \) in the matrix
\[ \left[\begin{array}{cccc} 4 & 2 & -2 & -2 \\ 0 & 3 & 0 & 0 \\ -2 & -4 & 7 & 4 \\ 4 & 8 & -8 & -5 \end{array}\right] \]
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 1 & 2 & -2 & -2 \\ 0 & 0 & 0 & 0 \\ -2 & -4 & 4 & 4 \\ 4 & 8 & -8 & -8 \end{array}\right] = \left[\begin{array}{cccc} 1 & 2 & -2 & -2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
A basis of the eigenspace is \( \left\{ \left[\begin{array}{c} -2 \\ 1 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} 2 \\ 0 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} 2 \\ 0 \\ 0 \\ 1 \end{array}\right] \right\} \).
Explain how to find a basis for the eigenspace associated to the eigenvalue \( -3 \) in the matrix
\[ \left[\begin{array}{cccc} -4 & 3 & -2 & 2 \\ -2 & 2 & -8 & -6 \\ -1 & -1 & -6 & -8 \\ -2 & 5 & -6 & -5 \end{array}\right] \]
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} -1 & 3 & -2 & 2 \\ -2 & 5 & -8 & -6 \\ -1 & -1 & -3 & -8 \\ -2 & 5 & -6 & -2 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
A basis of the eigenspace is \( \left\{ \left[\begin{array}{c} 0 \\ -2 \\ -2 \\ 1 \end{array}\right] \right\} \).
Explain how to find a basis for the eigenspace associated to the eigenvalue \( -2 \) in the matrix
\[ \left[\begin{array}{cccc} 5 & 7 & 7 & 7 \\ -3 & -5 & -3 & -3 \\ -3 & -3 & -5 & -3 \\ 5 & 5 & 5 & 3 \end{array}\right] \]
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 7 & 7 & 7 & 7 \\ -3 & -3 & -3 & -3 \\ -3 & -3 & -3 & -3 \\ 5 & 5 & 5 & 5 \end{array}\right] = \left[\begin{array}{cccc} 1 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
A basis of the eigenspace is \( \left\{ \left[\begin{array}{c} -1 \\ 1 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} -1 \\ 0 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -1 \\ 0 \\ 0 \\ 1 \end{array}\right] \right\} \).
Explain how to find a basis for the eigenspace associated to the eigenvalue \( 1 \) in the matrix
\[ \left[\begin{array}{cccc} 2 & 3 & 2 & -1 \\ 0 & 1 & 0 & 0 \\ -1 & -3 & -1 & 1 \\ 2 & 6 & 4 & -1 \end{array}\right] \]
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 1 & 3 & 2 & -1 \\ 0 & 0 & 0 & 0 \\ -1 & -3 & -2 & 1 \\ 2 & 6 & 4 & -2 \end{array}\right] = \left[\begin{array}{cccc} 1 & 3 & 2 & -1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
A basis of the eigenspace is \( \left\{ \left[\begin{array}{c} -3 \\ 1 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} -2 \\ 0 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} 1 \\ 0 \\ 0 \\ 1 \end{array}\right] \right\} \).
Explain how to find a basis for the eigenspace associated to the eigenvalue \( 3 \) in the matrix
\[ \left[\begin{array}{cccc} 6 & 3 & 3 & 3 \\ -1 & 2 & -1 & -1 \\ 5 & 5 & 8 & 5 \\ 3 & 3 & 3 & 6 \end{array}\right] \]
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 3 & 3 & 3 & 3 \\ -1 & -1 & -1 & -1 \\ 5 & 5 & 5 & 5 \\ 3 & 3 & 3 & 3 \end{array}\right] = \left[\begin{array}{cccc} 1 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
A basis of the eigenspace is \( \left\{ \left[\begin{array}{c} -1 \\ 1 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} -1 \\ 0 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -1 \\ 0 \\ 0 \\ 1 \end{array}\right] \right\} \).
Explain how to find a basis for the eigenspace associated to the eigenvalue \( 2 \) in the matrix
\[ \left[\begin{array}{cccc} 3 & -2 & -2 & 1 \\ -4 & 10 & 8 & -4 \\ 2 & -4 & -2 & 2 \\ 4 & -8 & -8 & 6 \end{array}\right] \]
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 1 & -2 & -2 & 1 \\ -4 & 8 & 8 & -4 \\ 2 & -4 & -4 & 2 \\ 4 & -8 & -8 & 4 \end{array}\right] = \left[\begin{array}{cccc} 1 & -2 & -2 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
A basis of the eigenspace is \( \left\{ \left[\begin{array}{c} 2 \\ 1 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} 2 \\ 0 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -1 \\ 0 \\ 0 \\ 1 \end{array}\right] \right\} \).
Explain how to find a basis for the eigenspace associated to the eigenvalue \( 4 \) in the matrix
\[ \left[\begin{array}{cccc} 5 & 1 & 1 & -2 \\ 0 & 5 & 3 & -1 \\ 0 & -1 & 1 & 1 \\ 1 & 1 & 1 & 2 \end{array}\right] \]
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 1 & 1 & 1 & -2 \\ 0 & 1 & 3 & -1 \\ 0 & -1 & -3 & 1 \\ 1 & 1 & 1 & -2 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -2 & -1 \\ 0 & 1 & 3 & -1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
A basis of the eigenspace is \( \left\{ \left[\begin{array}{c} 2 \\ -3 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} 1 \\ 1 \\ 0 \\ 1 \end{array}\right] \right\} \).
Explain how to find a basis for the eigenspace associated to the eigenvalue \( -1 \) in the matrix
\[ \left[\begin{array}{cccc} 0 & -7 & -8 & 2 \\ 0 & 0 & 1 & 0 \\ 2 & -4 & -7 & 4 \\ 1 & -5 & -6 & 1 \end{array}\right] \]
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 1 & -7 & -8 & 2 \\ 0 & 1 & 1 & 0 \\ 2 & -4 & -6 & 4 \\ 1 & -5 & -6 & 2 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -1 & 2 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
A basis of the eigenspace is \( \left\{ \left[\begin{array}{c} 1 \\ -1 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -2 \\ 0 \\ 0 \\ 1 \end{array}\right] \right\} \).
Explain how to find a basis for the eigenspace associated to the eigenvalue \( 4 \) in the matrix
\[ \left[\begin{array}{cccc} 5 & 1 & 2 & 1 \\ 0 & 5 & 4 & 5 \\ 0 & 0 & 5 & 1 \\ 0 & 1 & 5 & 10 \end{array}\right] \]
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 1 & 1 & 2 & 1 \\ 0 & 1 & 4 & 5 \\ 0 & 0 & 1 & 1 \\ 0 & 1 & 5 & 6 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & -2 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
A basis of the eigenspace is \( \left\{ \left[\begin{array}{c} 2 \\ -1 \\ -1 \\ 1 \end{array}\right] \right\} \).