(b): Let B be a fixed 2 x 2 real matrix. Let T : M2x2(R) → M2x2(R) be defined by T(A) = AB. Prove that if B is an invertible 2 x 2 matrix, then T is an invertible linear transformation. [Hint: Can you construct an inverse transformation U : M2x2(R) → M2x2(R) that takes advantage of the existence of B-l? Make sure that you verify that the U constructed really is an inverse of T.]
(b): Let B be a fixed 2 x 2 real matrix. Let T : M2x2(R) → M2x2(R) be defined by T(A) = AB. Prove that if B is an invertible 2 x 2 matrix, then T is an invertible linear transformation. [Hint: Can you construct an inverse transformation U : M2x2(R) → M2x2(R) that takes advantage of the existence of B-l? Make sure that you verify that the U constructed really is an inverse of T.]
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Part b from the picture below.
![(a): Suppose A1 and A2 are distinct eigenvalues of a linear operator T : V → V, with corresponding
eigenvectors v1 and v2, respectively. Prove that {V1, V2} is linearly independent.
(b): Let B be a fixed 2 x 2 real matrix. Let T: M2x2(R) → M2x2(R) be defined by
T(A) = AB.
Prove that if B is an invertible 2 x 2 matrix, then T is an invertible linear transformation. [Hint:
Can you construct an inverse transformation U : M2x2(R) → M2x2(R) that takes advantage of the
existence of B-1? Make sure that you verify that the U constructed really is an inverse of T.]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F27260fae-539c-4ca6-9fed-6022b8026087%2F43b145e1-d8ee-418b-9a59-42773f4e6cc4%2Fbks9wn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:(a): Suppose A1 and A2 are distinct eigenvalues of a linear operator T : V → V, with corresponding
eigenvectors v1 and v2, respectively. Prove that {V1, V2} is linearly independent.
(b): Let B be a fixed 2 x 2 real matrix. Let T: M2x2(R) → M2x2(R) be defined by
T(A) = AB.
Prove that if B is an invertible 2 x 2 matrix, then T is an invertible linear transformation. [Hint:
Can you construct an inverse transformation U : M2x2(R) → M2x2(R) that takes advantage of the
existence of B-1? Make sure that you verify that the U constructed really is an inverse of T.]
Expert Solution
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Step 1: Definitions used and given information.
Linear transformation: Let U and V are vector space over a field F. A mapping is linear transformation if it satisfies following conditions
a) .
b)
Remark 1: Let V be a finite dimensional vector space over a field F. Then a if one-to-one if and only if it is onto.
Remark 2: If f and g are two functions such that then f is invertible and .
Given: We have given a mapping defined by , where is invertible matrix.
Step 2: Show that T is invertible linear transformation.
We have given a mapping defined by , where is invertible matrix.
Let
a)
Consider,
b)
Consider,
From part a and b we see that T satisfies conditions of linear transformation hence T is linear transformation.
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