The _____________ is/are an important entry point for toxic gases, solvents, aerosols, and particulates.

When attempting to determine human toxicity to a specific toxicant, it is best to test using _______.

Please follow these instructions carefully, as failure to follow them will result in a penalty. (i) Download the Exam template from the class website and use that to write your solutions.  (ii) The solutions to each problem should be in the space specially assigned to them. (iii) When scanning your solutions into a pdf file, each page must be scanned as a separate page and the entire exam as one pdf file. (iv) You have 135 minutes to complete the exam, including the time to scan the exam and upload it as a pdf file to Proctorio   I.  Find all solutions to  the system of equations     x_1+ 3x_2+x_3+x_4=3   2x_1-2x_2+x_3+2x_4=8 3x_1+x_2+2x_3-x_4=-1                     (20 points) II.  Let A denote the coefficient matrix for the system of equations given in I.  i) Find a basis for the Column space of A.  (10 points) ii) Find a basis for the Row space of   (5 points) iii) Find a basis for the Null space of A (10 points) iv) What is the rank of A? What is the nullity of A? (5 points) III. Solve the matrix equation A.x=b, where A= ,  x= and b= by first finding the inverse of the matrix A.(25 points)   IV.  Write the matrix A in III as a product of elementary matrices. (25 points)

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.