MATH136 Lecture Notes - Lecture 10: Row Echelon Form, Hyperplane, Augmented Matrix

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ultramarineelephant806

26 Jul 2018

School

University of Waterloo

Department

Mathematics

Course

MATH136

Professor

Alejandra Vicente Colmenares

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Math 136 lecture 10 24 jan, 2018. [(cid:883)(cid:882)(cid:885) (cid:883)(cid:884)(cid:888) (cid:883) (cid:888) (cid:887) | (cid:883)(cid:884)(cid:886)](cid:2871) (cid:885)(cid:2869)[(cid:883)(cid:882)(cid:882) (cid:883)(cid:884)(cid:885) (cid:883) (cid:888) 8 | (cid:883)(cid:884)(cid:883)](cid:883)(cid:884)(cid:2870)[(cid:883)(cid:882)(cid:882) (cid:883)(cid:883)(cid:885) (cid:883) (cid:885) 8 | (cid:883)(cid:883)(cid:883)](cid:2869) (cid:2870) (cid:2871) (cid:885)(cid:2870) (cid:2870)+(cid:885)(cid:2871)[(cid:883)(cid:882)(cid:882) (cid:882)(cid:883)(cid:882) (cid:882)(cid:882)(cid:883) | 8 (cid:887) (cid:884)]=> (cid:2869)=8 (cid:2871)= (cid:884) =[8 (cid:887) (cid:884)] : other than substituting and/or eliminating (and also inclusive), we are. Definition: the act of applying elementary row operations to a matrix is called row reducing. Definition: two matrices are said to be row-equivalent if there exist a sequence of elementary row operations that converts one matrix into the other. If [ | ], and, [(cid:2869) |(cid:2869) ], are row equivalent, then they have the same solution set. (the converse of this statement is also true, because row operations are reversible) Examples: (cid:883)) [(cid:883)(cid:882)(cid:882) (cid:884)(cid:882)(cid:882) (cid:882)(cid:883)(cid:882) (cid:882) (cid:885)(cid:882) ] is in rref (cid:884)) [(cid:882)(cid:882)(cid:883) (cid:883)(cid:882)(cid:884) (cid:884)(cid:882)(cid:883) (cid:882)(cid:882)(cid:883) ] ~ [(cid:883)(cid:882)(cid:882) (cid:884)(cid:883)(cid:882) (cid:883) (cid:884)(cid:882) (cid:883)(cid:882)(cid:882) ] does not satisfy (condition 4)

Related Questions

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Q18. A size 6 dress in Country C is size 52 in Country D. A function that converts dress sizes in Country C to those in Country D is f(x) = 2(x + 20). Find a formula for the inverse of the function described.
a. f ^-1(x) = (x - 20)/2
b. f ^-1(x) = x - 20
c. f ^-1(x) = (x/2) - 20
d. f ^-1(x) = (x/2) + 20

Q19. If f(x) = 4^x and g(x) = 12, find the point of intersection of the graphs of f and g by solving f(x) = g(x). Give an exact answer.
a. (log ₄ 12, 12)
b. (log ₄ 12, 4)
c. (log ₄ 12, 0)
d. (12, 12)

Q20. Find functions f and g so that f â 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