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\centerline{Mathematics 244 -- Linear Algebra}
\centerline{Discussion 2 -- Change of Basis, Compositions}
\centerline{April 2, 2007; writeups due: April 4, 2007}
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{\it Background} 
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We have discussed the matrices of linear mappings and
change of basis in general.  In this discussion, we will
practice using these ideas.  We will also introduce compositions
of linear mappings and think about their matrices.
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{\it Discussion Questions}
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I.  Let $V = P_2({\bf R})$, let $W = M_{2\times 2}({\bf R})$, and 
consider the mapping $T : V \to W$ defined for $p \in P_2({\bf R})$ by 
$$T(p) = \pmatrix{p(0) & p(1)\cr
                 p'(0) & p'(1)\cr}.$$
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\item{A)}  Show that $T$ is a linear mapping.
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\item{B)}  Suppose we use the (ordered) bases $B = \{1,x,x^2\}$ in $V$ and
$C = \{e_{11},e_{12},e_{21},e_{22}\}$ in $W$.  What is the matrix 
$[T]_{C \leftarrow B}$?
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\item{C)}  Now suppose we change basis to $B' = \{1 - x^2, x + x^2, 2 + x^2\}$
for $P_2({\bf R})$.  What is the matrix $[T]_{C \leftarrow B'}$?
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\item{D)}  Show that $C' = \{e_{11},e_{22},e_{12}+e_{21},e_{12} - e_{21}\}$
is also a basis for $W$.
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\item{E)}  What is the matrix $[T]_{C' \leftarrow B'}$ ($B'$ same as in part C)? 
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II.  Let $V,W,U$ be three finite-dimensional vector spaces, and 
let $A,B,C$ be bases in $V,W,U$ respectively.
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\item{A)}  If $T : V \to W$ and $S : W \to U$ are linear mappings, 
show that the composition $S\circ T : V \to U$ 
(the mapping defined by $(S\circ T)(v) = S(T(v))$ for all $v \in V$)
is also a linear mapping.
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\item{B)}  For example, let $V = P_3({\bf R})$, let $W = P_4({\bf R})$, and 
let $U = P_4({\bf R})$.  Let $B = \{1,x,x^2,x^3,x^4\}$ and 
$A = C = \{1,x,x^2,x^3\}$ be bases.  
  Let $T: V \to W$ be the mapping
$T(p) = P$, where $P$ is the antiderivative of $p$ with $P(0) = 0$.
Let $S: W \to U$ be the mapping
$S(p) = p'$ (the derivative). 
What are the matrices 
$$[T]_{B\leftarrow A}, [S]_{C\leftarrow B}, [S\circ T]_{C\leftarrow A}$$
What theorem from calculus is reflected in your answer for the matrix
of $S\circ T$?
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\item{C)}  In general (not just in the case considered in part B), 
how are the matrices $[T]_{B\leftarrow A}$, $[S]_{C\leftarrow B}$
and $[S\circ T]_{C\leftarrow A}$ related?  
\item{D)} {\bf Extra Credit}  Prove your assertion in part C.
\bye
