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\title{
Linear Transformations}
\author{Silvius Klein}
\date{}
\vspace{2in}

\institute{MA1202 Linear Algebra with Applications}



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\begin{document}


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\begin{frame}
\frametitle{General linear transformations}

\underline{Definition}: Let $V$, $W$ be two vector spaces. A function $T \colon V \to W$ is called a {\em linear transformation} from $V$ to $W$ if the following hold for all vectors $u, v$ in $V$ and for all scalars $k$.
\begin{itemize}
\item[(i)] $T (u + v) = T (u) + T (v)$
(additivity)

\item[(ii)] $T (k u) = k T (u)$
(homogeneity)
\end{itemize}

\pause

If $V$ and $W$ are the same, we call a linear transformation from $V$ to $V$ a {\em linear operator}.

\pause

\medskip

\underline{Theorem}: A function $T \colon V \to W$  is a linear transformation if and only if for all vectors $v_1, v_2$ in $V$ and for all scalars $k_1, k_2$ we have
$$T (k_1 \, v_1 + k_2 \, v_2) = k_1 \, T (v_1) + k_2 \, T (v_2)$$


\end{frame}


\begin{frame}
\frametitle{General linear transformations} 



\underline{Theorem} (basic properties of linear transformations): If $T$ is a linear transformation then
\begin{itemize}
\item[a)] $T (\vec{0}) = \vec{0}$

\item[b)] $T ( - v) = - T (v)$

\item[c)] $T (u - v) = T (u) - T (v)$
\end{itemize}

\pause
\medskip

\underline{Theorem}: If $T \colon V \to W$  is a linear transformation, $S = \{ v_1, v_2, \ldots, v_n \}$ is a basis in $V$, then for any vector $v$ in $V$ we can evaluate $T (v)$ by
$$T (v) = c_1 \, T (v_1) + c_2 \, T (v_2) + \ldots + c_n \, T (v_n)$$
where $v = c_1 \, v_1 + c_2 \, v_2 + \ldots + c_n \, v_n$.
\end{frame}


\begin{frame}
\frametitle{Kernel and range of a linear transformation}

\underline{Definition}: Let $T \colon V \to W$  is a linear transformation. 
\begin{itemize}

\item The set of all vectors $v$ in $V$ for which $T (v) = \vec{0}$ is called the {\em kernel} of $T$.

We denote the kernel of $T$ by $\ker (T)$.

\item The set of all outputs (images) $T (v)$ of vectors in $V$ via the transformation $T$ is called the {\em range} of $T$.

We denote the range of $T$ by $R (T)$. 

\end{itemize}

\pause
\medskip

\underline{Theorem}: If $T \colon V \to W$  is a linear transformation, then $\ker (T)$ is a {\em subspace} of $V$, while $R (T)$ is a subspace of $W$.

\pause
\medskip

\underline{Definition}: If $V$ and $W$ are {\em finite} dimensional vector spaces and $T \colon V \to W$  is a linear transformation, then we call
\begin{itemize}
\item $\dim \ker (T) = \text{ nullity of } T$
\item $\dim R (T) = \text{ rank of } T $
\end{itemize}

\pause
\medskip

\underline{Theorem}: If $V$ and $W$ are finite dimensional vector spaces and $T \colon V \to W$  is a linear transformation, then
$$\text{ rank } (T) + \text{ nullity } (T) = \dim (V)$$

\end{frame}


\begin{frame}
\frametitle{One-to-one and onto functions}

\underline{Definition} (one-to-one function): A function $f \colon X \to Y$ is called {\em one-to-one} if to distinct inputs it assigns distinct outputs. 

More precisely, $f$ is 1-1 means: if $x_1 \neq x_2$ then $f (x_1) \neq f (x_2)$. This is logically equivalent to saying that if $f (x_1) = f (x_2)$ then $x_1 = x_2$.

\pause

\bigskip

\underline{Definition} (onto function): A function $f \colon X \to Y$ is called {\em onto} if every element in $Y$ is an output of $f$. 

More precisely, $f$ is onto if for every $y$ in $Y$ there is at least one $x$ in $X$ such that $f (x) = y$.

\pause

\bigskip

Linear transformations are functions, so being one-to-one or onto applies (makes sense) for them as well.


\end{frame}

\begin{frame}
\frametitle{Isomorphism}

\underline{Theorem}: A linear transformation  $T \colon V \to W$  is one-to-one if and only if $\ker (T) = \{ \vec{0} \}$.

\medskip

\underline{Theorem}: Let  $T \colon V \to V$ be a linear operator, where $V$ is a \alert{finite} dimensional vector space.

The following statements are equivalent.
\begin{itemize}
\item[a)] $T$ is one-to-one
\item[b)] $\ker (T) = \{ \vec{0} \}$
\item[c)] $T$ is onto.
\end{itemize}

\medskip

\underline{Definition}: A linear transformation $T \colon V \to W$ which is one-to-one and onto is called an {\em isomorphism}. 

Two vector spaces $V$ and $W$ are called {\em isomorphic} if there is an isomorphism  $T \colon V \to W$.

\medskip

\underline{Examples}: $P_{n-1}$ is isomorphic to $\R^n$. $M_{2 \times 2} (\R)$ is isomorphic to $\R^4$.

\medskip

\underline{Theorem}: Every $n$ dimensional vector space is isomorphic to $\R^n$.
\end{frame}

\begin{frame}
\frametitle{The matrix for a linear transformation: definition}

We have:
\begin{itemize} 
\item[$\diamond$] An $n$-dimensional vector space $V$ with a basis $B = \{ u_1, u_2, \ldots, u_n \}$. 
\item[$\diamond$] An $m$-dimensional vector space $W$ with a basis $B'$.
\item[$\diamond$]  A linear transformation  $T \colon V \to W$.
\end{itemize}

\smallskip

\underline{Definition}: The {\em matrix for $T$ relative to the bases $B$ and $B'$} is the $m \times n$ matrix $[ T ]_{B', B}$ defined by

$$ [ T ]_{B', B} = \big[  \, [ T (u_1) ]_{B'}  \, | \,  [ T (u_2) ]_{B'} \, |  \ldots | \,[ T (u_n) ]_{B'}  \big] $$

\medskip

Relative to these bases, we can think of the linear transformation $T$ as simply the multiplication transformation from $\R^n$ to $\R^m$ by the matrix $[ T ]_{B', B}$. More precisely, we have the following relation:

$$ \big[ T (x) \big]_{B'} = [ T ]_{B', B} \cdot [ x ]_B$$


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\begin{frame}
\frametitle{The matrix for a linear transformation: properties}

In class we have learned the following.

\medskip

\underline{Theorem}: If  $T \colon V \to V$ is a linear operator and if $B$ is a basis for $V$, then the following are equivalent:
\begin{itemize} 
\item[(a)] $T$ is one-to-one. 
\item[(b)] $[ T ]_{B, B}$ is invertible.
\end{itemize}

Moreover, if these conditions hold, then

$$ [ T^{- 1} ]_{B, B} = [ T ]_{B, B}^{- 1}$$

\medskip

But in fact much more is true: given any linear transformation $T$ from a vector space $V$ to another vector space $W$, the matrix for $T$ in two chosen basis $[ T ]_{B', B}$ completely encodes whether the transformation is one-to-one, onto or an isomorphism.

\end{frame}


\begin{frame}
\frametitle{The matrix for a linear transformation: more properties}

Let $V$ be an $n$-dimensional vector space with a basis $B$ and let $W$ be an $m$-dimensonal vector space with a basis $B'$. 
Let $T \colon V \to W$ be a linear transformation.

\underline{Theorem}: The following are equivalent:
\begin{itemize} 
\item[(a)] $T$ is one-to-one. 
\item[(b)] The null space of $[ T ]_{B, B}$ is $\{ \vec{0} \}$.
\item[(c)] ${\rm nullity} \, [ T ]_{B', B} = 0$.
\end{itemize}

\smallskip

\underline{Theorem}: The following are equivalent:
\begin{itemize} 
\item[(a)] $T$ is onto. 
\item[(b)] The column space of $[ T ]_{B, B}$ is $\R^m$.
\item[(c)] ${\rm rank} \, [ T ]_{B', B} = \dim (W)$.
\end{itemize}

\smallskip





\underline{Theorem}: The following are equivalent:
\begin{itemize} 
\item[(a)] $T$ is an isomorphism 
\item[(b)] $\dim (V) = \dim(W)$ and $[ T ]_{B', B}$ is invertible.
\end{itemize}






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