1. (a) Find the matrix P that multiplies every vector (x, y, z) E R³ to produce the vector (3x+2y+z, 5y-z, 8x). Also find P-1.(b) Find the matrix P that multiplies every vector (x, y) E R2 to produce (5x-4y, - 2x, 3y - 2x) = R³

Answered: 1. (a) Find the matrix P that…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Related questions

Question

1. (a) Find the matrix P that multiplies every vector (x, y, z) E R³ to produce the vector (3x+2y+z, 5y-
z, 8x). Also find P-1.
(b) Find the matrix P that multiplies every vector (x, y) E R2 to produce (5x-4y, - 2x, 3y - 2x) = R³

Expert Solution

Step 1

What is Linear Transformation:

A function from one vector space to another that respects the underlying (linear) structure of each vector space is called a linear transformation. A linear operator, or map, is another name for a linear transformation. The transformation is known as an endomorphism or, if invertible, an automorphism when the range of the transformation matches the domain. The underlying field of the two vector spaces must be identical.

To Determine:

We determine the matrix $P$ which over multiplication with all vectors $(x, y, z) \in ℝ^{3}$ produces $(3 x + 2 y + z, 5 y - z, 8 x) \in ℝ^{3}$ . We also determine $P^{- 1}$ .
We determine the matrix $P$ which over multiplication with all vectors $(x, y) \in ℝ^{2}$ produces $(5 x - 4 y, - 2 x, 3 y - 2 x) \in ℝ^{3}$ .