1. Let S = {[][ED (a) Show that S is a basis for R³. (b) (i) Find a matrix Q that converts from S to the standard basis for R³. (ii) Find a matrix P that converts from the standard basis in R³ to S. (c) Let f: R→ R such that f(x, y, z) = (2x - 3y, 2y3z, 2z - 3x). Show that f is a linear transformation. (i) (ii) Find a matrix A representing f relative to the standard basis in R³. (iii) Use your answers to part (b) to find a matrix B representing f relative to the basis S.
1. Let S = {[][ED (a) Show that S is a basis for R³. (b) (i) Find a matrix Q that converts from S to the standard basis for R³. (ii) Find a matrix P that converts from the standard basis in R³ to S. (c) Let f: R→ R such that f(x, y, z) = (2x - 3y, 2y3z, 2z - 3x). Show that f is a linear transformation. (i) (ii) Find a matrix A representing f relative to the standard basis in R³. (iii) Use your answers to part (b) to find a matrix B representing f relative to the basis S.
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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![1. Let S = {[], [¹], [2]}.
(a) Show that S is a basis for R³.
(b) (i)
(ii)
Find a matrix Q that converts from S to the standard basis for R³.
Find a matrix P that converts from the standard basis in R³ to S.
(c) Let f: R→R such that f(x, y, z) = (2x − 3y, 2y —- 3z, 2z - 3x).
Show that f is a linear transformation.
(i)
(ii)
Find a matrix A representing f relative to the standard basis in R³.
(iii)
Use your answers to part (b) to find a matrix B representing f relative to the basis
S.
(iv) What effect does f have on area and orientation?
(d) Let A be a 3x3 matrix. Show that det(A) = 0 if and only if the columns of A are
linearly dependent.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd63740e0-8dd1-40ff-ae15-48815d78b54a%2F1aa891d5-acae-45e1-8b77-be3ac2ac56f2%2F3veedy7_processed.png&w=3840&q=75)
Transcribed Image Text:1. Let S = {[], [¹], [2]}.
(a) Show that S is a basis for R³.
(b) (i)
(ii)
Find a matrix Q that converts from S to the standard basis for R³.
Find a matrix P that converts from the standard basis in R³ to S.
(c) Let f: R→R such that f(x, y, z) = (2x − 3y, 2y —- 3z, 2z - 3x).
Show that f is a linear transformation.
(i)
(ii)
Find a matrix A representing f relative to the standard basis in R³.
(iii)
Use your answers to part (b) to find a matrix B representing f relative to the basis
S.
(iv) What effect does f have on area and orientation?
(d) Let A be a 3x3 matrix. Show that det(A) = 0 if and only if the columns of A are
linearly dependent.
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