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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.  Solve the  matrix equation: .    (10 points)   II. Here you need to prove that the rank of the matrix A= is 2.   (10 points)

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 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)

List the four chordate characteristics.

Match the following organisms with the correct descriptions.

Radially symmetrical aquatic invertebrates with stinging cells located on tentacles belong to which phylum?

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

What fungal group does the fungi picture below belong to?  

Characteristics shared by both snakes and birds include: