| # Exercises Linear Equations > Exercises ## Nullspace, rank and range Determine the nullspace, range and rank of a matrix of the form where , with , and . In the above, the zeroes are in fact matrices of zeroes with appropriate sizes. Consider the matrix with , . What is the size of ? Determine the nullspace, the range, and the rank of . ## Dimension of solution set Determine the dimension of the sets of solutions to the following linear equations in a vector . Hint: use dyads and matrix notation. ## Solving linear equations with multiple right-hand sides Often it is of interest to be able to solve linear equations of the form for many different instances of the output vector . In this problem we assume that we are given such instances , , which are collected as columns of a matrix . A direct approach to this task is encapsulated in the following matlab snippet:
    n = 500; p = 100; A = randn(n); B = randn(n,p);
    tic
    for k=1:p
    	b = B(:,k);
    	x = A\b;
    end
    fprintf(’elapsed time = %10.7f\n’, toc)
Here is another approach:
    n = 500; p = 100; A = randn(n); B = randn(n,p);
    tic
    [Q,R] = qr(A);
    for k=1:p
    	b = B(:,k);
    	x = R\(Q’*b);
    end
    fprintf(’elapsed time = %10.7f\n’, toc)
Which method is faster? Justify your answer. ## Polynomial interpolation In this problem, we look at a simple application of the range space for fitting a polynomial through a set of points. We are given points in and we want to find a polynomial of degree such that, for all , we have . That is, the polynomial must go through all the points. A polynomial of degree is parametrized by the vector of its coefficients, that is: We assume that if . What is the smallest value of that ensures that we will be able to fit any points? Would your answer be the same without the assumption that if ? Explain briefly why the assumption is important or does not change the answer. We are given a set of points. How can we compute the smallest value of such that there exists a polynomial that goes through all the points? How would you compute the coefficients of the polynomial? We are told that the points were drawn from a polynomial of degree and that up to one point is faulty (the polynomial does not go through this point). Is there a faulty point (discuss depending on the respective value of and )? If yes, how can you find it?