MATH 220 Midterm: MATH220 Washington State Math220MidSum16

#4. Questions II. and III.

1) solving the following system of linear equations:

x1+ 2x2 + 3x3 + 4x4 + 5x5= 6

x1+ 2x2 + 4x3 + 3x4 + 7x5= 5

x1+ 2x2 + 2x3 + 5x4 + 4x5= 9

Write the solution in parametric form.

2) Define a function T:R^(3) to R^(2) by T(x,y,z) = (x + y + z,x +2y - 3z)

a) Show that T is a linear transofrmation

b) Find all the vectors in the kernel of T

c) Show that T is onto

d) Find the matrix representation of T relative to the standardbasis of R^(3) and R^(2)

3) Show that B= {(1,1,1), (1,1,0), (0,1,1)} is a basis forR^(3). Find the coordinate vector of (1,2,3) relative to the basisof B.

4) Use the Gram-Schmidt process to find an orthonormal basis forthe subspace of R^(4) spanned by the vectors(1,0,1,1), (1,0,1,0), (0,0,1,1).

5) Consider the Matrix B = {(3,2,-3), (-3,-4,9), (-1,-2,5)}.Find an invertible matrix P and a diagonal matrix D such thatB=PDP^(-1).

ALL WORK AND STEPS MUST BE SHOWN OR YOUR RESPONSE WILL NOT GETRATED!!

Q20. Find functions f and g so that f ∘ g = H(x) = (5 - 2x³)².
a. f(x) = 5 - 2x³ ; g(x) = x²
b. f(x) = (5 - 2x)³ ; g(x) = x²
c. f(x) = x² ; g(x) = 5 - 2x³
d. f(x) = x³ ; g(x) = (5 - 2x)²

Q29. Solve the system of equations using Cramer's Rule if it is applicable.


a. x = 9, y = 2, z = 5; (9, 2, 5)
b. x = 8, y = 4, z = 5; (8, 4, 5)
c. x = 4, y = 5, z = 4; (4, 5, 4)
d. x = 8, y = -4, z = -5; (8, -4, -5)

Q30. Solve the system of equations using matrices (row operations). If the system has no solution, say that it is inconsistent.


a. x = 2, y = 3, z = -7; (2, 3, -7)
b. x = 1, y = 3, z = -4; (1, 3, -4)
c. x = 5, y = 13/2, z = -3; (5, 13/2, -3)
d. x = 1, y = -3, z = -16; (1, -3, -16)

Q33. Write the partial fraction decomposition of the rational expression (4x³ + 4x²)/(x² + 5)².
a. (4x - 4)/(x² + 5) + (-20x + 20)/(x² + 5)²
b. (4x + 4)/(x² + 5) + (-20x - 20)/(x² + 5)²
c. (4x + 4)/(x² + 5) + (20x - 20)/(x² + 5)²
d. (4x + 4)/(x² + 5) + (20x + 20)/(x² + 5)²

Q34. Solve the linear programming problem. Minimize z = 11x + 6y + 7 subject to: x = 0, y = 0, x + y = 1.
a. minimum: 7
b. minimum: 18
c. minimum: 24
d. minimum: 13

Q35. Solve the system of equations by elimination.


a. x = 10, y = 0; (10, 0)
b. x = 10, y = 10; (10, 10)
c. x = 0, y = 10; (0, 10)
d. x = 0, y = 0; (0, 0)

Q36. Perform the row operations on the given augmented matrix.


a.
b.
c.
d.

Q37. Solve the system of equations using Cramer's Rule if it is applicable.


a. x = -4, y = 3; (-4, 3)
b. x = -3, y = -4; (-3, -4)
c. x = 4, y = 3; (4, 3)
d. x = 3, y = 4; (3, 4)

Q38. Write the partial fraction decomposition of the rational expression x/(x² - 9x + 20).
a. -4/(x - 4) + 5/(x - 5)
b. -5/(x - 4) + 4/(x - 5)
c. 4/(x - 4) + -5/(x - 5)
d. -4/(x - 4) + -5/(x - 5)

Q39. Solve the system of equations.


a. x = -4, y = 4, z = -3; (-4, 4, -3)
b. x = -3, y = 4, z = -4; (-3, 4, -4)
c. x = -4, y = -3, z = 4; (-4, -3, 4)
d. inconsistent

Q40. Find the value of the determinant.


a. 224
b. 24
c. -72
d. 72