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\noindent
{\Large Math 211 \hfill Homework 10 \hfill Sample Texfile}

\medskip
\noindent{\bf 1.} (10 points)
Show that 
$${\mathbb B} = \left\{1, 1-t, 2-4t+t^2, 6-18t+9t^2-t^3\right\}$$
is a basis for $\mathbb{P}_3$, and find the change of basis matrix from 
${\mathbb B} $ to ${\mathbb C} = \{1, t, t^2, t^3\}$ and the change of the basis 
matrix from ${\mathbb C} $ to ${\mathbb B}$
Find $[u]_{\mathbb B}$ for $u = 1 - 3t + 9t^2 -t^3$.


\medskip
\noindent{\bf 2.} (5 points)
 Find the rank, the nullity (dimension of null space), a basis for the 
column space, a basis for the row space of the following matrix
$$\left[\matrix{1&1&-2&0&1&0\cr 0&1&-1&0&-3&0\cr 0&0&1&1&-13&0\cr 
0&0&0&0&1&0\cr 0&0&0&0&0&1}\right].$$


\medskip\noindent
{\bf 3.}
Find the characteristic polynomial for the matrix and two linearly independent 
eigenvectors for the matrix 
$$A=\left[\matrix{4 & 4\cr 1&4}\right].
$$

\medskip\noindent
{\bf 4.} Show that $A = \left[\matrix{1 & 2 & 0 \cr 0 & 1 & 3 \cr 0 & 0 & 1}\right]$
has at most one linear independent eigenvector.

\medskip
\noindent{\bf 5.}
Find the characteristic polynomial for the matrix and three linearly independent 
eigenvectors for the matrix 
$$A=\left[\matrix{4&0&-1\cr 0&3&4\cr 0&0&2}\right].
$$
 
\medskip
\noindent{\bf 6.}
Find the values $h$ so that  there are two linearly independent
eigenvectors corresponding to the eigenvalue $\lambda =4$
 for the matrix
$$A=\left[\matrix{4&2&3&3\cr 0&2&h&3\cr 0&0&4&14\cr 0&0&0&2}\right].$$

[Hint: Find $h$ such that the null space of $A-4I_4$ has nullity 2.]


\medskip\noindent
{\bf 7.} Show that $A$ and $A^T$ have the same characteristic polynomial.

[Hint: Show that $\det(A- \lambda I) = \det(A^T - \lambda I)$.]

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