3 5. (Representing Subspaces As Solutions Sets of Homogeneous Linear Systems; the problem requires familiarity with the full text of the material entitled "Subspaces: Sums and Intersections" on the course page). Let 1 2 -2 2 5 -2 -2 -3 2 -0-0-0-0-0-0 -1 -1 5 and = -1 -14 -28 35 14 27 -31, 2 1 let L₁ Span(₁, 2, 3), = and let L₂ = Span(d₁, d₂, da). (i) Form the matrix C1 C = ₂¹ whose rows are the transposed column vectors ci. (a) Take the matrix C to reduced row echelon form; (b) Use (a) to find a basis for L₁ and the dimension dim(L₁) of L₁; (c) Use (b) to find a homogeneous line ar system S₁ whose solution set is equal to L₁. (ii) Likewise, form the matrix di D = d₂¹ d3¹ (iii) (a) Find the general solution of the combined linear system S1 U S2; (b) use (a) to find a basis for the intersection L₁ L₂ and the dimension of the intersection LL₂3 (c) use (b) to find the dimension of the sum L₁ + L₂ of L₁ and L₂. Present your answers to the problem in a table of the following form Answers Subproblem (i) (a) Reduced row echelon form of the matrix C; (b) Basis for L₁, the dimension of L₁; (c) Homogeneous linear system S₁. (ii) (a) Reduced row echelon form of the matrix D; (b) Basis for L2, the dimension of L2; (c) Homogeneous line ar system S₂. (iii) (a) General solution of the system S₁ U S2; (b) Basis for L₁ L₂ (c) Dimension of L₁ + L2.