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\title
Math 261 -- Linear Algebra\vskip 10pt
Fall 1999 (Exam 2)
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\noindent (1) Recall that $P_n$ denotes the space of real polynomials 
of degree at most
$n$ and consider the linear transformation
$T:P_n\to P_{n+1}$ given by
$(Tf)(x)=\int_0^xf(t)dt$.
Write down the matrix of $T$ relative to the canonical bases of $P_n$ 
and $P_{n+1}$.
\vskip 10pt

\noindent (2) Let $T:V\to V$ be a linear operator, where $V$ is 
finite-dimensional. A subspace
$W$ of $V$ is said to be
$T$ -- \it invariant \rm  if $TW \subset W$. Start with a basis for 
$W$ and then complete it to a
basis $\beta$ of $V$. What does the matrix of $T$ relative to $\beta$
look like?

\vskip10pt

\noindent (3) Give bases for the kernel and the range of the linear map $T:\Bbb
R^3\to \Bbb R^4$ given by
$$T(x_1, x_2, x_3)= (3x_1 +x_2-4x_3, 2x_2 +x_3, 3x_1-2x_3, x_2-x_3).$$
\vskip10pt

\noindent (4) Consider the subspaces $W_1$ and $W_2$ of $\Bbb 
R^{13}$, $\dim W_1=8$, $\dim
W_2=7$.  What are the possible dimensions of $W_1\cap W_2$? Explain 
your answer carefully and
give examples to illustrate some of the possibilities.
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\noindent (5) a) Define what it means for two vector spaces to be 
isomorphic. b) Exhibit
an isomorphism between $\Bbb R^{n+1}$ and the space of polynomials of 
degree at most $n$ .
c) Prove that if two finite-dimensional vector spaces are isomorphic 
then they have the same
dimension. d) Does the converse of c) hold ?  You must settle d) by 
either giving a proof
or by exhibiting a counter-example.
\vskip10pt

\noindent (6) Write an essay about products of matrices and how they 
relate to composition of
linear transformations. Your essay must include examples and the 
proof of at least one important
theorem. You can borrow responsibly from known sources but the 
examples should be your own.

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