1. (Use format short) Consider the set of functions 1,cost, cos2 t,cos t) We can view this set as a set of vectors in the vector space C(R) of all continuous real-valued functions whose domain in R. We would like to show that this is a linearly independent set in C(R). This means that if 1,2,3,4 are scalars such that (*) ?(1) +x2 cos t + z3 cos2 t + z4 cosa t = 0 (for all t), then we must have z1-0? 0,23-0,24-0. Essentiany we are trying to disprove the existence of a lincar relation relating these four functions to cach other. More concretely, we are trying to prove that no identity such as could possibly exist. (a) Each substitution of a number for t in the equation ) produces a linear equation in the four variables zi,,, T. By plugging in t-0,0.1,0.2 and 0.3, you get four linear equations for the four unknowns. Define the coefficient matrix A for this linear system in MATLAB (b) Note that a nontrivial solution x to () is automatically a nontrivial solution to Ax 0. However if A is invertible, then A 0 has no nontrivial solutions. This implies that the equation (*) has no nontrivial solutions. Compute rref (A) and det (A) (c) Very briefly explain why each of the last two computations show A is invertible. (d) It is rensonable to be suspicious of the very small value of det(A) in the last step could this be roundoff error, with the actual det (A) being zero? Do a check by repeating the computation with the more spread out inputs t-0,.2,.5,1, to see det(A) large enough to eliminate that suspicion. (e) Now consider the set of functions (1,sin2 t, cos2 t). (Here 1 denotes the constant function whose value is always 1.) Explain why this set of functions is linearly dependent. (Hint: Do you know any identities that relate these three functions?) (f) (Optional don't have to turn in) Explore what happens if you try to use the techniques of the previous parts on the set (1,sin2 t,cos2 t) 3 -27 1 2. (Use format rat) Let A4 283 101 0 (a) Compute rank A. (The MATLAB command is rank(A).) (b) Use the rank to determine the values dim(Nul A), dim(Col A), and dim(Row A). (c) Compute rref (A) and use it to give a basis for i. Nul A ii. Col A iii. RowA PLEASE USE MATLAB TO SOLVE AND SHOW WORK THANKS