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{\Large  18.099 - 18.06CI.} 

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{\large HW-3} 
 

{Due on Monday, March 8 in class. 
First draft due on Thursday, Feb 26. First \LaTeX draft due 
on Thursday, March 4.}

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\begin{enumerate}

\item{Let $L$ and $M$ be finite dimensional linear spaces and let 
$f: L \to M$ be a linear map. Prove that the subspaces $\ker f$ and 
${\rm Im }f$ are finite dimensional and 
$$ \dim \ker f + \dim {\rm Im}f = \dim L .$$ 
Hint: Use the extension of a basis theorem.}

\item{
Consider the space ${\mathcal L}(V,V)$ 
of linear maps 
from a finite 
dimensional real linear space $V$ to itself.
Which of the following subsets of 
${\mathcal L}(V,V)$ 
are linear spaces: 
\begin{enumerate} 
 \item{ $\{f \in {\mathcal L}(V,V) \; | \;\; \dim {\rm Im}f =0\};$ }
 \item{ $\{f \in {\mathcal L}(V,V) \; | \;\; \dim {\ker}f =0\};$ }
 \item{ $\{f \in {\mathcal L}(V,V) \; | \;\; \dim {\rm Im}f < \dim V\}.$} 
\end{enumerate} 
Reformulate the obtained results in terms of matrices.}

\item{\begin{enumerate} 
      \item{Show that there are no finite square real or complex matrices 
$X$ and $Y$ such that $XY -YX =I$, where $I$ is the 
identity matrix. }
      \item{Check that multiplication by $x$ and differentiation 
$\frac{d}{dx}$ are linear maps on the (real or complex) 
space $\mathcal P$ of all polynomials in one variable $x$. 
Compute the action of 
$\frac{d}{dx} \circ x - x \circ \frac{d}{dx}$ in $\mathcal P$. 
Here ``$\circ$'' denotes the composition of two maps. } 
      \item{Introduce the concept of a matrix with infinitely many  
rows and columns. Define the multiplication for infinite 
matrices in which each column and each row has a finite 
number of non-zero elements. Using (b), find a solution 
for $XY -YX =I$ in terms of infinite matrices.} 
\end{enumerate} }

\end{enumerate} 


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