To prove: That for every integer n ≥ 2 the number of lines obtained by joining n distinct points in the plane, no three of which are collinear, is 1 2 n ( n − 1 ) .

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Chapter 5.1, Problem 27E

To determine

To prove: That for every integer n≥2 the number of lines obtained by joining n distinct points in the plane, no three of which are collinear, is 12n(n−1).

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Page < 2 of 2 - ZOOM + The set of all 3 x 3 upper triangular matrices 6) Determine whether each of the following sets, together with the standard operations, is a vector space. If it is, then simply write 'Vector space'. You do not have to prove all ten vector space axioms. If it is not, then identify one of the ten vector space axioms with its number in the attached sheet that fails and also show that how it fails. a) The set of all polynomials of degree four or less. b) The set of all 2 x 2 singular matrices. c) The set {(x, y) : x ≥ 0, y is a real number}. d) C[0,1], the set of all continuous functions defined on the interval [0,1]. 7) Given u = (-2,1,1) and v = (4,2,0) are two vectors in R³-space. Find u xv and show that it is orthogonal to both u and v. 8) a) Find the equation of the least squares regression line for the data points below. (-2,0), (0,2), (2,2) b) Graph the points and the line that you found from a) on the same Cartesian coordinate plane.

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Page < 1 of 2 - ZOOM + 1) a) Find a matrix P such that PT AP orthogonally diagonalizes the following matrix A. = [{² 1] A = b) Verify that PT AP gives the correct diagonal form. 2 01 -2 3 2) Given the following matrices A = -1 0 1] an and B = 0 1 -3 2 find the following matrices: a) (AB) b) (BA)T 3) Find the inverse of the following matrix A using Gauss-Jordan elimination or adjoint of the matrix and check the correctness of your answer (Hint: AA¯¹ = I). [1 1 1 A = 3 5 4 L3 6 5 4) Solve the following system of linear equations using any one of Cramer's Rule, Gaussian Elimination, Gauss-Jordan Elimination or Inverse Matrix methods and check the correctness of your answer. 4x-y-z=1 2x + 2y + 3z = 10 5x-2y-2z = -1 5) a) Describe the zero vector and the additive inverse of a vector in the vector space, M3,3. b) Determine if the following set S is a subspace of M3,3 with the standard operations. Show all appropriate supporting work.

Answer 2

Question

Chapter 5.1, Problem 27E

To determine

To prove: That for every integer n≥2 the number of lines obtained by joining n distinct points in the plane, no three of which are collinear, is 12n(n−1).

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Answer 3

Question

Chapter 5.1, Problem 27E

To determine

To prove: That for every integer n≥2 the number of lines obtained by joining n distinct points in the plane, no three of which are collinear, is 12n(n−1).

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Answer 4

Question

Chapter 5.1, Problem 27E

To determine

To prove: That for every integer n≥2 the number of lines obtained by joining n distinct points in the plane, no three of which are collinear, is 12n(n−1).

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Answer 5

Chapter 5.1, Problem 27E

To determine

To prove: That for every integer n≥2 the number of lines obtained by joining n distinct points in the plane, no three of which are collinear, is 12n(n−1).

Answer 6

Chapter 5.1, Problem 27E

To determine

To prove: That for every integer n≥2 the number of lines obtained by joining n distinct points in the plane, no three of which are collinear, is 12n(n−1).

Answer 7

Chapter 5.1, Problem 27E

To determine

To prove: That for every integer n≥2 the number of lines obtained by joining n distinct points in the plane, no three of which are collinear, is 12n(n−1).

Answer 8

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Answer 9

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Answer 10

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Answer 11

Ch. 5.1 - True/False Questions The statement i=1n(2i1)=n2...Ch. 5.1 - Prob. 2TFQ Ch. 5.1 - Prob. 3TFQ Ch. 5.1 - Prob. 4TFQ Ch. 5.1 - Prob. 5TFQ Ch. 5.1 - Prob. 6TFQ Ch. 5.1 - Prob. 7TFQ Ch. 5.1 - Prob. 8TFQ Ch. 5.1 - Prob. 9TFQ Ch. 5.1 - Prob. 10TFQ

To prove: That for every integer n ≥ 2 the number of lines obtained by joining n distinct points in the plane, no three of which are collinear, is 1 2 n ( n − 1 ) .

Solution Summary: The author proves that the number of lines obtained by joining n distinct points in the plane, no three of which are collinear, is 12n(n-1)

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To prove: That for every integer n≥2 the number of lines obtained by joining n distinct points in the plane, no three of which are collinear, is 12n(n−1).

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