Matrix | Tensor Analysis Answers

This question applies the ideas of linear algebra to fitting a graphs to data.
(a) Suppose I want to find a quadratic equation of the form y = a + bx + cx2 to pass through the points (-2,25), (3,0) and (6,33). Explain how this is related to the matrix equation [[1,-2,4],[1,3,9],[1,6,36]]*[a,b,c]=[25,0,33], and hence use matrix techniques to find a, b and c. Interpret your solution in terms of writing [25,0,33] as a linear combination of the vectors that form the columns of the matrix.
(b) Write down the matrix equation A[a,b,c]=v to solve if I also want the quadratic equation to go
through (5,10). Express the fact that this is not possible in terms of a vector not being a linear
combination of the vectors that form the columns of a matrix.
(c) Explain why, if v = u + w where u is a linear combination of the columns of A and w is orthogonal
to each of the columns of A, then A^T v = A^Tu, and hence deduce that if [a0,b0,c0] is a solution of
A^TA[a,b,c]=A^T v, then A[a0,b0,c0]=u and the error w is small.

1. Find the matrix that represents the linear transformation of the plane obtained by reflecting in the line y=x, then rotating anticlockwise through an angle of 45 degrees, and finally reflecting in the y-axis. Give a simpler geometrical description of what this transformation does.

Factorize the denominator of f(x)=(2x^5+15x^4+15x^3+2x^2+2)/(x^5+2x^4+x^3-x^2-2x-1) completely, and use this to write f(x) as a partial fraction.

Find the eigenvalues and eigenvectors of the symmetric matrix S = [[-1,2,0],[2,2,1],[0,1,-1]]. Let M be the 3x3 matrix whose columns are the eigenvectors you have found. Evaluate M^T*M, with as little computation as possible. Give reasons for any computations you were able to omit.

Quadrilateral BURT has vertices B(6,1), U(3,5), R(-1,4), and T (-3,-5). What translation matrix would you need to use to translate BURT so that R' has coordinates (3,2)?