Problem 1 Consider the following system of 2 linear equations and 3 variables: 3r2 - Gra = 9 TI - 2x2 | 3ra = 4 (a) (Ipt) TRUE or FALSE: is a solution to this system. (b) (3 pts) Use elementary row operations to calculate the reduced row echelon form of the augmented matrix of this system. You must label all row operations performed (c) (2 pts) Which are the free variables? Which are the leading variables? (d) (I pt) Calculate the rank of the coefficient matrix of this system. (e) (3 pts) Calculate all the solutions to this system. Write your answers using vector notation.Problem 2 Let L be the line in R given by the equation 21 1 3x2 = 0, let projy : R- > R be the orthogonal projection onto L, and let refz : R- > R be the reflection on L. Let cy = n and &2 -"- (a) (1 pt) Calculate a vector of perpendicular to L. (b) (4 pts) Calculate the orthogonal projections projz (e ) and projz(62) of the vectors , and 62 onto L. (c) (2 pts) Calculate a matrix, A, such that proj, (F) = ME. (d) (2 pts) Calculate a matrix, B, such that ref ,.(F) = BE. (e) (I pt) Use (d) to calculate the reflection of |on L.Problem 3 (a) (2 pts) Which of the following matrices represent a rotation of R' about the origin? 3 0.6 0.8 0.6 0.6 0.8 0.6 0.8 0.8 0.6 0.6 0.8 (b) (2 pts) Let T' : R. > R" be the linear transformation given by T" = I1 - 2:2 | 3ca Which is the matrix A such that 7(F) = As for all vectors a? 3 3 2 2 0 I -3 0 3 2 (c) (2 pts) Calculate the inverse, A-, of the matrix 4 = 3 2 5 7 5 -2 3 -7 (d) (2 pts) Let A = 2 and B = P 1 2. Calculate the product BA. O BA= 3 0 2 6 O BA= 3 4 3 U BA= 2 -2 4 U BA= 0 -2 12 7 11 (e) (2 pts) Which of the following statements are always true for all linear transformations T' : R" -> R"? IT(0) =0. I T(KE) = T(k) + T(#), for all scalars k and all vectors a in R.". I I There exists an n x m matrix 1 such that T'(@) = As, for all a in R.". I I There exists an w x n matrix 4 such that T'(@) = Ar, for all & in R"