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.)

Answer problem I on page 2 of the Exam template. Answer the remaining problems, one on each page. IMPORTANT: Please return the entire 8 pages of the exam even if you write only on a few of the 8 pages. I. Answer the following questions by just writing T (True) or F (False) only. (3 points each) i) if A is an m x n-matrix so that A* x=  0, forevery vector  x in  R^n, then A is the zero-matrix. ii) If A and B are nonsingular matrices, then so is A+B.iii) If A and B are nonsingular matrices, then so is A*B.iv) Suppose A is an n x n-matrix so that A^10=I . Then0 is not an eigenvalue for A.v) Suppose A is an nxn matrix so that A^10 =I. Then det(A) cannot be zero. vi). Suppose A is an nxn matrix so that det(A) =0. Then A has 0 as an eigenvalue. vii). Let A denote a 6×9 matrix. Then dim N(A) =3.viii) Let A denote a 9 x 6 matrix with Rank(A) =6. Then dim(N(A^T)) =3.The next two questions refer to the following situation.Let S = { v_1, …,  v_k} be k non-zero vectorsin R^n.ix). If V = Span (S) and dim (V) = k, S is a basis for V.x). If v_1,…, v_{k-1} are linearly independent, thenso is S.   II. Let A= .  a) Find all the eigen-values of A.  (10 points) b) Find the corresponding eigen-vectors.  (10 points) c) Find a basis for R3 with respect to which the corresponding linear transformation can be diagonalized.  (5 points) d) Find the corresponding diagonal matrix. (5 points)   III.  Let  A = .(a) Find a basis for the null-space of A. (15 points)(b) Find a basis for the column-space of A (15 points)(c) Let P_4 be the set of polynomials in one variable t and of degree

Answer problem I on page 2 of the Exam template. Answer the remaining problems, one on each page. IMPORTANT: Please return the entire 8 pages of the exam even if you write only on a few of the 8 pages. I. Answer the following questions by just writing T (True) or F (False) only. (3 points each) i) if A is an m x n-matrix so that A* x=  0, forevery vector  x in  R^n, then A is the zero-matrix. ii) If A and B are nonsingular matrices, then so is A+B.iii) If A and B are nonsingular matrices, then so is A*B.iv) Suppose A is an n x n-matrix so that A^10=I . Then0 is not an eigenvalue for A.v) Suppose A is an nxn matrix so that A^10 =I. Then det(A) cannot be zero. vi). Suppose A is an nxn matrix so that det(A) =0. Then A has 0 as an eigenvalue. vii). Let A denote a 6×9 matrix. Then dim N(A) =3.viii) Let A denote a 9 x 6 matrix with Rank(A) =6. Then dim(N(A^T)) =3.The next two questions refer to the following situation.Let S = { v_1, …,  v_k} be k non-zero vectorsin R^n.ix). If V = Span (S) and dim (V) = k, S is a basis for V.x). If v_1,…, v_{k-1} are linearly independent, thenso is S.   II. Let A= .  a) Find all the eigen-values of A.  (10 points) b) Find the corresponding eigen-vectors.  (10 points) c) Find a basis for R3 with respect to which the corresponding linear transformation can be diagonalized.  (5 points) d) Find the corresponding diagonal matrix. (5 points)   III.  Let  A = .(a) Find a basis for the null-space of A. (15 points)(b) Find a basis for the column-space of A (15 points)(c) Let P_4 be the set of polynomials in one variable t and of degree

Please answer questions 1 and 2 on page 1 and questions 3 and 4 on page 2. Let A= .  1. Find all the eigenvalues of A.   (7 points) 2. Find the corresponding eigen vectors.   (7 points) 3. Can A be diagonalized?   (2 points) 4. If the answer to 3 is yes, find a matrix X and a diagonal matrix D so that X^{-1}*A*X is a diagonal matrix D. (4 points) Old Quiz 6: ————- 1.  Use Cramer’s rule to find the solutions of the system of equations: x1+ 3×2+ x3=1 2×1+x2+x3  =5 -2×1+2×2 – x3= -8   This can broken into the following subproblems: (i) Compute the determinant of the coefficient matrix for the system of equations above.  (5 points) (ii) Compute the determinant of the three matrices obtained by replacing one column of the coefficient matrix by the vector on the right hand side.  (4 points each) (iii) Write the solutions for each of the variables x1, x2 and x3. (3 points)    

Your answer to the following question should consist of a single word. An interspecific interaction in which both species benefit is known as: 

Bonus Question: 1 pt What was your favorite part of the class?

Predation of a rat by a snake can be categorized as an ______ interaction.

Penguins that live in the South Pole must deal with the freezing temperature and cold winds. What type of ecological factor is this?

Sponges are different from all other animals in that they lack true _____. (Your answer should consist of a single word.)

The arachnids (spiders, scorpions, ticks) are the most successful arthropod group.