Explanation Given: The angle is θ ( 0 ≤ θ < 2 π ) and θ ≠ π 2 , 3 π 2 . Approach: In transformation of ℝ 2 with invertible matrix can be obtained by applying an appropriate sequence of reflection, shear and stretches and represented as, T ( v ) = A v The transformation M 1 ( k ) = [ k 0 0 1 ] corresponds to the linear stretch in the x -axis. The transformation A 12 ( k ) = [ 1 0 k 1 ] corresponds to shear line parallel to y -axis. The relation between sine, cosine and tangent is, tan θ = sin θ cos θ The Pythagorean identity is, sin 2 θ + cos 2 θ = 1 The relation between cosine and secant is, sec θ = 1 cos θ Calculation: The matrix that corresponds to a rotation through an angle θ in the counterclockwise direction is, A = [ cos θ − sin θ sin θ cos θ ] Apply the row operation M 1 ( 1 cos θ ) in the matrix A . A = [ 1 − sin θ cos θ sin θ cos θ ] Consider the relation between sine, cosine and tangent. A = [ 1 − tan θ sin θ cos θ ] Apply the row operation A 12 ( − sin θ ) . A = [ 1 − tan θ 0 cos θ − ( − sin θ ) ( − tan θ ) ] Consider the relation between sine, cosine and tangent. A = [ 1 − tan θ 0 cos θ − ( − sin θ ) ( − sin θ cos θ ) ] = [ 1 − tan θ 0 cos θ + ( − sin θ ) ( − sin θ cos θ ) ] = [ 1 − tan θ 0 cos 2 θ + sin 2 θ cos θ ] Consider the relation between cosine and secant and the Pythagorean identity. A = [ 1 − tan θ 0 sec θ ] Apply the row operation M 2 ( 1 sec θ ) . A = [ 1 − tan θ 0 1 ] Apply the row operation A 21 ( tan θ )

Differential Equations And Linear Algebra, Books A La Carte Edition (4th Edition)

4th Edition

ISBN: 9780321985811

Author: Stephen W. Goode, Scott A. Annin

Publisher: Pearson (edition 4)

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Chapter 6.2, Problem 14P

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To verify:

The matrix of the transformation and the same in terms of reflections, stretches and shears.

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Chapter 6.2, Problem 13P

Chapter 6.3, Problem 1TFR

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موضوع الدرس Prove that Determine the following groups Homz(QZ) Hom = (Q13,Z) Homz(Q), Hom/z/nZ, Qt for neN- (2) Every factor group of adivisible group is divisble. • If R is a Skew ficald (aring with identity and each non Zero element is invertible then every R-module is free.

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Chapter 6 Solutions

Differential Equations And Linear Algebra, Books A La Carte Edition (4th Edition)

Ch. 6.1 - True-False Review For Questions a-f, decide if the...Ch. 6.1 - True-False Review For Questions a-f, decide if the...Ch. 6.1 - True-False Review For Questions a-f, decide if the...Ch. 6.1 - True-False Review For Questions a-f, decide if the...Ch. 6.1 - True-False Review For Questions a-f, decide if the...Ch. 6.1 - True-False Review For Questions a-f, decide if the...Ch. 6.1 - For problem 1-8, verify directly from Definition...Ch. 6.1 - For problems 1-8, verify directly from Definition...Ch. 6.1 - For problem 1-8, verify directly from Definition...Ch. 6.1 - For problem 1-8, verify directly from Definition...

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To verify: The matrix of the transformation and the same in terms of reflections, stretches and shears.

To verify: The matrix of the transformation and the same in terms of reflections, stretches and shears.

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Chapter 6 Solutions

To verify:

The matrix of the transformation and the same in terms of reflections, stretches and shears.