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 \(a_{i,j} = \left|\begin{matrix}x_{i,1}&x_{i,2}\\x_{j,1} &x_{j,2}\end{matrix}\right|\). This is a determinant of a matrix composed from the i-th and the j-th rows of x. I remind that \(\left|\begin{matrix}a&b\\c&d\end{matrix}\right|=ad - bc\).

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 \(a_1x+b_1y+c_1\) = 0 and \(a_2x+b_2y+c_2 = 0\):

    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.