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Q43E

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Linear Algebra With Applications
Found in: Page 74
Linear Algebra With Applications

Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

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Short Answer

Use the formula derived in Exercise 2.1.13 to find the inverse of the rotation matrix

localid="1659346816315" A=[cosθ-sinθsinθcosθ].

Interpret the linear transformation defined by A-1geometrically. Explain.

The inverse of the matrix is, A-1=cosθsinθ-sinθcosθ , the geometrical interpretation is,

.

See the step by step solution

Step by Step Solution

Step 1: Compute the inverse of the matrix.

Consider the matrix.

A=cosθ-sinθsinθcosθ

The inverse of the matrix is,

A-1=1cosθ×cosθ-sinθ×-sinθcosθsinθ-sinθcosθA-1=1cos2θ+sin2θcosθsinθ-sinθcosθA-1=cosθsinθ-sinθcosθ

Step 2: Interpret the linear transformation.

The matrix A represents the rotation by an angle θ and scaling by a factor r.

The inverse matrix A-1 represents the rotation by an angle -θ and scaling by a factor 1r.The geometrical representation is,

Step 3: Final answer.

The inverse matrix is, A-1=cosθsinθ-sinθcosθ with geometrical interpretation of linear

transformation is,

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