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{\Huge 189-251B: Algebra 2} \\ \vskip 0.1in
{\Huge Assignment 2} \\ \vskip 0.1in
{\Large Due: Wednesday, January 22 }
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1. Let $\C$ be the field of complex numbers and let
$V$ be the $\C$-vector space $\C^3$.
Find the coordinates of the vector $(1,0,1)$ in the basis
$(v_1,v_2,v_3)$, where
$$ v_1=(2i,1,0), v_2=(2,-1,1), v_3 =(0,1+i,1-i).$$
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2. Let $V$ denote the $\Q$-vector space of $2\times 2$ matrices with entries in
$\Q$.
Let $A$ be the matrix $\mat{2}{-5}{3}{17}$. Show that the function
$$T:V\lra V \quad \mbox{ defined by } T(X) = AX - XA$$
is a linear transformation from $V$ to $V$.
Choose a basis for $V$, and write down the matrix of $T$ with respect to your basis.
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3. Let $V$ be a finite-dimensional vector space.
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a. Show that any
linear transformation $T:V\lra V$ is injective if and only if it is surjective.
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b. Show that neither property implies the other, if the finite-dimensionality
assumption on $V$ is dropped.
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4. Let $V$ and $W$ be finite-dimensional vector spaces, of dimension $n$ and $m$
respectively, over the field $\Z_2$.
How many functions are there from $V$ to $W$? How many linear transformations?
What are the dimensions of these spaces of functions, and linear transformations, respectively?
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5. Let $V$ be the set of real-valued sequences $(a_n)_{n\ge 0}$ satisfying
$$ a_{n+1} = a_n + a_{n-1} \quad \mbox{ for all } n\ge 1.$$
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a.
Show that $V$, equipped with the usual internal addition and scalar
multiplication on sequences, is a vector space over $\R$.
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b.
What is the dimension of $V$ over $\R$?
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c. A {\em geometric progression } is a sequence of the
form $a_n= \alpha^n$,
for some $\alpha\in\R$.
Show that $V$ has a basis consisting of geometric progressions, by producing
such a basis.
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d.
The {\em Fibonacci sequence} is the unique sequence $(a_n)\in V$ satisfying
$$a_0=a_1=1, \quad \mbox{ so that } \quad a_2=2, a_3=3, a_4=5, a_5=8, a_6=13, \quad \mbox{etc.} $$
Express this sequence as a linear combination of the basis vectors obtained
in part c. Deduce from this a closed form expression for the $n$th term of the Fibonacci
sequence.
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