Name:

1. Write the row echelon form $𝑬$ of $𝑨$ by observation, without carrying out any calculations

   $𝑨$	$𝑬$
   $\left[\begin{array}{lll}1& 2& 3\\ 2& 4& 6\\ 3& 6& 9\end{array}\right]$	$\left[\begin{array}{lll}& & \\ & & \\ & & \end{array}\right]$
   $\left[\begin{array}{lll}2& 0& 1\\ 1& 0& \frac{1}{2}\\ 0& 4& 0\end{array}\right]$	$\left[\begin{array}{lll}& & \\ & & \\ & & \end{array}\right]$
   $\left[\begin{array}{lll}0& 1& 0\\ 0& 0& 2\\ 3& 0& 0\end{array}\right]$	$\left[\begin{array}{lll}& & \\ & & \\ & & \end{array}\right]$

2. For $𝒖={\left[\begin{array}{ll}2& 1\end{array}\right]}^T$, $𝒗={\left[\begin{array}{ll}-1& 3\end{array}\right]}^T$, compute the following

   1. ${𝒖}^T𝒗$

   2. $𝒖{𝒗}^T$

3. For $f:[-1,1]\to ℝ$, $f$ differentiable an infinite number of times

   1. Write the MacLaurin series of $f$ (Taylor series around $x_0=0$)

   2. What is the codomain of $f$?

   3. What is the domain of $f$?

4. For $f:[-\pi ,\pi ]\times [-\pi ,\pi ]\to ℝ$, $f(x,y)=\sin x\cos y$ differentiable an infinite number of times

   1. Write the MacLaurin series of $f$ (Taylor series around $x_0=0$)

   2. What is the gradient of $f$?

5. Compute the products showing your calculation procedure

   ${\displaystyle 𝑪=𝑨𝑩=\left[\begin{array}{llll}1& 0& -2& 1\\ 0& 1& 1& 2\\ -1& 2& 1& 0\\ 2& 1& 0& 1\end{array}\right]\left[\begin{array}{llll}1& 0& -2& 1\\ 0& 1& 1& 2\\ -1& 2& 1& 0\\ 2& 1& 0& 1\end{array}\right],𝑫=𝑩𝑨.}$

   Write pseudo-code that evaluates a polynomial through Horner's scheme

   ${\displaystyle p_n\left(x\right)=(((a_{n}x+a_{n-1})x+a_{n-2})x+\cdots +a_1)x+a_0}$

6. Write code in some language (e.g., Python, C, ) to implement Horner's scheme.

7. After class, use ChatGPT to obtain an implementation of Horner's scheme.