I. Given the matrix A= ,  write A as a product of elementary…

I. Given the matrix A= ,  write A as a product of elementary matrices (10 points)   II. For the matrix B= , determine the LU factorization. (10 points, 5 for L, 5 for U)   III.  Which of the following are subspaces of R3 ? (a) V= { v = | x+y +z=0}            (b) W= { w= | x+2y+3z=1}   (10 points, 5 points each.)

I.  (i) Given vectors v_1= , and v_2 =,  determine their spa…

I.  (i) Given vectors v_1= , and v_2 =,  determine their span as a plane in 3-space.  (10 points)  (ii) Is the vector x =  in the span of v_1 and v_2? Give reasons for your answer. (5 points) II.  Given the matrix A= ,  (i) find an echelon form for it. (5 points)     (ii) Determine the null-space of A, using the echelon form in (i).  (5 points)     (iii) Determine a basis for the null-space of A. (5 points)

I.  (i) Given vectors v_1= , and v_2 =,  determine their spa…

I.  (i) Given vectors v_1= , and v_2 =,  determine their span as a plane in 3-space.  (10 points)  (ii) Is the vector x = in the span of v_1 and v_2? Give reasons for your answer. (5 points) II.  Given the matrix A= ,  (i) find an echelon form for it. (5 points)     (ii) Determine the null-space of A, using the echelon form in (i).  (5 points)     (iii) Determine a basis for the null-space of A. (5 points)