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{\Huge 189-251B: Algebra 2} \\ \vskip 0.1in
{\Huge Assignment 3} \\ \vskip 0.1in
{\Large Due: Wednesday, January 29. }
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\noindent
1. Solve the linear equation $Ax=y$ over the field $\Z_2$,
where
$$ A = \left( \begin{array}{ccc}
0 & 1 & 1 \\
1 & 0 & 1 \\
1 & 1 & 1
\end{array} \right), \quad y = \left(\begin{array}{c} 0 \\ 1 \\ 1 \end{array} \right).$$
\sk\noindent
2. Solve the linear equation $Ax=y$ over the field $\Z_2$,
$$ A = \left( \begin{array}{ccc}
0 & 1 & 1 \\
1 & 0 & 1 \\
1 & 1 & 0
\end{array} \right), \quad y = \left(\begin{array}{c} 0 \\ 1 \\ 1 \end{array} \right).$$
How many solutions does this equation have? Write them all down.
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3. Show that the number of distinct solutions of a system of linear equations (in any
number of equations, and unknowns) over
the field $\Z_p$ is either $0$, or a power of $p$.
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\noindent
4. Show that there is no error-correcting code of dimension $5$ in
$\Z_2^7$, so that the example constructed in class is in some sense optimal,
in the sense that it is the error-correcting code of largest possible
dimension in $\Z_2^7$.
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\noindent
5. What is the largest possible dimension of a linear code in
$\Z_2^{15}$ that can detect and correct a single (i.e., one-bit) error?
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6. Let $V$ be the vector space of polynomials of degree at most
$5$ over the field
$\R$ of real numbers, and
let $T$ be the linear transformation from $V$ to $V$ defined by
$$T(f) = \frac{d^3}{dx^3} f + \frac{d^2}{dx^2} f. $$
Describe the kernel of $T$, and its image.
What are the dimensions of these subspaces?
What is the subspace of $V$ generated by $\ker(T)$
and Image$(T)$?
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7. A linear transformation $T:V\lra V$ is called a {\em projection}, or
an {\em idempotent}, if it satisfies $T^2=T$. Show that
$V$ can be expressed as the {\em direct sum} of
$\ker(T)$ and Image$(T)$.
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8. Let $V$ be a vector space over a field $F$.
A linear transformation $N:V\lra V$ is
said to be {\em nilpotent } if $N^k=0$ for
some $k$. Show that if $N$ is nilpotent, then the linear transformation
$1-N$ (where $1$ denotes the identity transformation) is
invertible. (Hint: think of the power series expansion for
$\frac{1}{1-x}$ when $x<1$.)
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