\magnification 1800
\parskip 10pt
\parindent 0pt
\hoffset -0.3truein
\hsize 7truein


\def\R{{\bf R}}
\def\C{{\bf C}}
\def\N{{\bf N}}
\def\r{{\bf r}}
\def\c{{\bf c}}
\def\p{{\bf p}}
\def\x{{\bf x}}
\def\y{{\bf y}}
\def\e{{\bf e}}
\def\i{{\bf i}}
\def\j{{\bf j}}
\def\k{{\bf k}}
\def\0{{\bf 0}}
\def\a{{\bf a}}
\def\emph{}

\def\S{\Sigma_{i=1}^n}
\def\Sm{\Sigma_{i=1}^m}

\def\s{{\sigma}}
\def\ep{\epsilon}
\def\f{\vskip 10pt}
\def\h{\hskip 10pt}
\def\u{\vskip -8pt}
\def\bh{\hfil \break}



Thm:  Let $T: V \rightarrow W$ be a linear transformation.  Then
$T({\bf 0}) = {\bf 0}$

Pf:  $T({\bf 0}) = T({\bf 0} +  {\bf 0}) = T({\bf 0}) +  T({\bf 0})$

\vskip 5pt
\hrule

Thm:  Let $A$ be an $m \times n$ matrix.  Then the function
$$\matrix{ T:\R^n \rightarrow \R^m \cr 
T({\bf x}) = A{\bf x}}$$ is a linear transformation.

Thm:  If $T: \R^n \rightarrow \R^m$ is a linear transformation,
then $T({\bf x}) = A{\bf x}$ where $$A = [T({\bf e_1}) ... T({\bf e_n})]$$

Ex:  If $T: \R^2 \rightarrow \R^2$ and

 $T\left( \matrix{1 \cr 0} \right) = \left( \matrix{2 \cr 3} \right)$, $T\left( \matrix{0 \cr 1} \right) =
\left( \matrix{1 \cr 4} \right)$, 
then 

$$T\left( \matrix{x \cr y} \right) = xT\left( \matrix{1 \cr 0} \right) + yT\left( \matrix{0 
\cr 1} \right) =
  x \left( \matrix{2 \cr 3} \right) + y \left( \matrix{1 \cr 4} \right) = \left( \matrix{2 
& 1 \cr 3 & 4} \right) \left(\matrix{x \cr y} \right)$$

Change of basis:

Suppose ${\cal S} = \{\left( \matrix{1 \cr 0} \right), \left( \matrix{0 \cr 1} \right) \}$

Suppose ${\cal B} = \{\left( \matrix{2 \cr 3} \right), \left( \matrix{1 \cr 4} \right) \}$

Defn: $\left( \matrix{x \cr y} \right)_{\cal B} = x{\bf b_1} + y {\bf b_2}$

Thus $\left( \matrix{1 \cr 0} \right)_{\cal B} = \left( \matrix{2 \cr 3} \right)_{\cal S}$
and $\left( \matrix{0 \cr 1} \right)_{\cal B} = \left( \matrix{1 \cr 4} \right)_{\cal S}$



$\left( \matrix{x \cr y} \right)_{\cal B} =
x\left( \matrix{2 \cr 3} \right)_{\cal S} + y\left( 
\matrix{1
\cr 4}
\right)_{\cal S} = \left( \matrix{2x + y \cr 3x + 4y} \right)_{\cal S}$

$\left( \matrix{2
& 1 \cr 3 & 4} \right) \left(\matrix{x \cr y} \right)_{\cal B} = \left(\matrix{u \cr 
v}\right)_{\cal S}$

Suppose ${\cal C} = \{\left( \matrix{5 \cr 2} \right), \left( \matrix{2 \cr 1} \right) \}$

$\left( \matrix{5
& 2 \cr 2 & 1} \right) \left(\matrix{w \cr z} \right)_{\cal C} = \left(\matrix{u \cr
v}\right)_{\cal S}$


$\left( \matrix{1
& -2 \cr -2 & 5} \right) \left(\matrix{u \cr v} \right)_{\cal S} = \left(\matrix{w \cr
z}\right)_{\cal C}$
\vskip 5pt
\hrule

$\left( \matrix{1
& -2 \cr -2 & 5} \right) \left( \matrix{2
& 1 \cr 3 & 4} \right) \left(\matrix{x \cr y} \right)_{\cal B} = 
\left( \matrix{1
& -2 \cr -2 & 5} \right) \left(\matrix{u \cr
v}\right)_{\cal S} =  \left(\matrix{w \cr
z}\right)_{\cal C}$

$\left( \matrix{-4
& -7 \cr 11 & 18} \right) \left(\matrix{x \cr y} \right)_{\cal B} =\left(\matrix{w \cr
z}\right)_{\cal C}$

Note: 
\vskip -10pt
$\left( \matrix{-4
& -7 \cr 11 & 18} \right) \left(\matrix{1 \cr 0} \right)_{\cal B} 
=\left(\matrix{-4 \cr 11}\right)_{\cal C}$
and $\left( \matrix{-4
& -7 \cr 11 & 18} \right) \left(\matrix{0 \cr 1} \right)_{\cal B} 
=\left(\matrix{-7 \cr 18}\right)_{\cal C}$


Take a chart  $(U, \phi)$  at $p$ where $\phi(p) = \0$,

The {\it standard  basis} for $T_p(M)$ w.r.t.$(U, \phi)$ =
$\{v_1, ...,
v_m\}$,

where  $v_i = D_{\alpha_i}$ and

 $\alpha_i: (-\ep, \ep)
\rightarrow M$,
$\alpha_i(t) = \phi^{-1}(0, ..., t, ..., 0) $  for some $\ep > 0$.

If $v \in T_p(M)$, then $v = \Sm a_i v_i$ where $a_i = v([\pi_i \circ \phi])$



Suppose $B x_{\cal B} = x_{\cal S}$ and $C x_{\cal C} = x_{\cal D}$

If $Tx_{\cal B} = y_{\cal C}$, then 
 $TB^{-1}B x_{\cal B} = C^{-1}C y_{\cal C}$

 $CTB^{-1} (B x_{\cal B}) = C y_{\cal C}$

 $CTB^{-1} x_{\cal S} = y_{\cal D}$




\end