Part b from the picture below. (a): Suppose A1 and A2 are distinct eigenvalues of a linear operator T : V → V, with correspondingeigenvectors 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 byT(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 theexistence of B-1? Make sure that you verify that the U constructed really is an inverse of T.]

Answered: (b): Let B be a fixed 2 x 2 real…

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter5: Orthogonality

Section5.4: Orthogonal Diagonalization Of Symmetric Matrices

Problem 27EQ

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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.]

Expert Solution

Step 1: Definitions used and given information.

Linear transformation: Let U and V are vector space over a field F. A mapping $T : V \to U$ is linear transformation if it satisfies following conditions

a) $T (x + y) = T (x) + T (y), \forall x, y \in V$ .

b) $T (α x) = α T (x), \forall x \in V, \forall α \in F .$

Remark 1: Let V be a finite dimensional vector space over a field F. Then a $T : V \to V$ if one-to-one if and only if it is onto.

Remark 2: If f and g are two functions such that $f \circ g = I = g \circ f$ then f is invertible and $f^{- 1} = g$ .

Given: We have given a mapping $T : M_{2 \times 2} (R) \to M_{2 \times 2} (R)$ defined by $T (A) = A B$ , where $B$ is invertible matrix.

Step 2: Show that T is invertible linear transformation.

We have given a mapping $T : M_{2 \times 2} (R) \to M_{2 \times 2} (R)$ defined by $T (A) = A B$ , where $B$ is invertible matrix.

Let $A, C \in M_{2 \times 2} (R), α \in F .$

Consider,

$T (A + C) = (A + C) B = A B + C B = T (A) + T (C) \Rightarrow T (A + C) = T (A) + T (C)$

Consider,

$T (α A) = (α A) B = α (A B) = α T (A) \Rightarrow T (α A) = α T (A)$

From part a and b we see that T satisfies conditions of linear transformation hence T is linear transformation.

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