477 \end{frame}

   478 

   479 \section{Solving linear equations}

   480 

   481 \begin{frame}[fragile]

   482 \frametitle{Solution of equations}

   483 Consider,

   484   \begin{align*}

   485     3x + 2y - z  & = 1 \\

   486     2x - 2y + 4z  & = -2 \\

   487     -x + \frac{1}{2}y -z & = 0

   488   \end{align*}

   489 Solution:

   490   \begin{align*}

   491     x & = 1 \\

   492     y & = -2 \\

   493     z & = -2

   494   \end{align*}

   495 \end{frame}

   496 

   497 \begin{frame}[fragile]

   498 \frametitle{Solving using Matrices}

   499 Let us now look at how to solve this using \kwrd{matrices}

   500   \begin{lstlisting}

   501     In []: A = array([[3,2,-1],

   502                       [2,-2,4],                   

   503                       [-1, 0.5, -1]])

   504     In []: b = array([[1], [-2], [0]])

   505     In []: x = solve(A, b)

   506     In []: Ax = dot(A,x)

   507   \end{lstlisting}

   508 \end{frame}

   509 

   510 \begin{frame}[fragile]

   511 \frametitle{Solution:}

   512 \begin{lstlisting}

   513 In []: x

   514 Out[]: 

   515 array([[ 1.],

   516        [-2.],

   517        [-2.]])

   518 \end{lstlisting}

   519 \end{frame}

   520 

   521 \begin{frame}[fragile]

   522 \frametitle{Let's check!}

   523 \begin{lstlisting}

   524 In []: Ax

   525 Out[]: 

   526 array([[  1.00000000e+00],

   527        [ -2.00000000e+00],

   528        [  2.22044605e-16]])

   529 \end{lstlisting}

   530 \begin{block}{}

   531 The last term in the matrix is actually \alert{0}!\\

   532 We can use \kwrd{allclose()} to check.

   533 \end{block}

   534 \begin{lstlisting}

   535 In []: allclose(Ax, b)

   536 Out[]: True

   537 \end{lstlisting}

   538 \inctime{15}

   539 \end{frame}

   540 

   541 \subsection{Exercises}

   542 

   543 \begin{frame}[fragile]

   544 \frametitle{Problem 1}

   545 Given the matrix:\\

   546 \begin{center}

   547 $\begin{bmatrix}

   548 -2 & 2 & 3\\

   549  2 & 1 & 6\\

   550 -1 &-2 & 0\\

   551 \end{bmatrix}$

   552 \end{center}

   553 Find:

   554 \begin{itemize}

   555   \item[i] Transpose

   556   \item[ii]Inverse

   557   \item[iii]Determinant

   558   \item[iv] Eigenvalues and Eigen vectors

   559   \item[v] Singular Value decomposition

   560 \end{itemize}

   561 \end{frame}

   562 

   563 \begin{frame}[fragile]

   564 \frametitle{Problem 2}

   565 Given 

   566 \begin{center}

   567 A = 

   568 $\begin{bmatrix}

   569 -3 & 1 & 5 \\

   570 1 & 0 & -2 \\

   571 5 & -2 & 4 \\

   572 \end{bmatrix}$

   573 , B = 

   574 $\begin{bmatrix}

   575 0 & 9 & -12 \\

   576 -9 & 0 & 20 \\

   577 12 & -20 & 0 \\

   578 \end{bmatrix}$

   579 \end{center}

   580 Find:

   581 \begin{itemize}

   582   \item[i] Sum of A and B

   583   \item[ii]Elementwise Product of A and B

   584   \item[iii] Matrix product of A and B

   585 \end{itemize}

   586 \end{frame}

   587 

   588 \begin{frame}[fragile]

   589 \frametitle{Solution}

   590 Sum: 

   591 $\begin{bmatrix}

   592 -3 & 10 & 7 \\

   593 -8 & 0 & 18 \\

   594 17 & -22 & 4 \\

   595 \end{bmatrix}$

   596 ,\\ Elementwise Product:

   597 $\begin{bmatrix}

   598 0 & 9 & -60 \\

   599 -9 & 0 & -40 \\

   600 60 & 40 & 0 \\

   601 \end{bmatrix}$

   602 ,\\ Matrix product:

   603 $\begin{bmatrix}

   604 51 & -127 & 56 \\

   605 -24 & 49 & -12 \\

   606 66 & -35 & -100 \\

   607 \end{bmatrix}$

   608 \end{frame}

   609 

   610 \begin{frame}[fragile]

   611 \frametitle{Problem 3}

   612 Solve the set of equations:

   613 \begin{align*}

   614   x + y + 2z -w & = 3\\

   615   2x + 5y - z - 9w & = -3\\

   616   2x + y -z + 3w & = -11 \\

   617   x - 3y + 2z + 7w & = -5\\

   618 \end{align*}

   619 \inctime{10}

   620 \end{frame}

   621 

   622 \begin{frame}[fragile]

   623 \frametitle{Solution}

   624 Use \kwrd{solve()}

   625 \begin{align*}

   626   x & = -5\\

   627   y & = 2\\

   628   z & = 3\\

   629   w & = 0\\

   630 \end{align*}