Vectorization

1. task7 You are given a row matrix, for example, a = [1, 2, -2, 4] and a number x, for example, 1. The row a specifies coefficients of a polinomial, starting from the lowest degree, that is, the row from the example specifies the polynomial $1+2x-2x^2+4x^3$. Compute the value of the polinomial at the point x. The answer in the example is 5.
2. task8 You are given a column matrix $x$. Create a square matrix of the same size, its value in the row number $i$ and the column number $j$ should be equal to $a_{i,j}=x_i–x_j$.
3. task9 You are given a three columns matrix. Imagine, that each row specifies a line of the form $ax+by+c=0$, whith the columns being, correspondingly, a, b, c. Return a matrix, that has two columns, correspondingly, k и b, that define the same lines, but in the form $y=kx+b$. For example, the row [1 1 1] should be converted to the row [-1 -1].
4. all2dets(x). You are given a matrix x of two columns. Return a new square matrix with the size equal to the number of rows in x. An element in the row number $i$ and the column number $j$ should be equal to $a_{i,j}=\left|\begin{array}{cc}{x}_{i,1}& x_{i,2}\\ x_{j,1}& x_{j,2}\end{array}\right|$. This is a determinant of a matrix composed from the i-th and the j-th rows of x. I remind that $\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|=ad-bc$.

5. all_lines_intersections(a). You are given a matrix а of three columns, it describes a set of lines (as previously). The short task statement is: intersect each line with each line. To be more specific: the intersection of the line from the i-th row and the line from j-th row has two coordinates $x$ and $y$. The function should return two matrices x and y. The first one contains the x-coordinate of this intersection in the row $i$ and the column $j$. The second one contains, correspondingly, the y-coordinate. Don’t consider the case of parallel lines.

   Here is the formula for intersection of lines $a_{1}x+b_{1}y+c_1$ = 0 and $a_{2}x+b_{2}y+c_2=0$:

    Δ = det([a1 b1; a2 b2])
    Δx = det([-c1 b1; -c2 b2])
    Δy = det([a1 -c1; a2 -c2])
    x = Δx / Δ
    y = Δy / Δ
   

   So, you will need a previously implemented all2dets function.