Let A =  a b c d , B =  e f g h ∈ M2,2(R) and define hA, Bi = ae + bf + 2cg + 2dh. (a) Show hA, Bi is an inner product of M2,2(R).

Question

Let A =  a b c d , B =  e f g h ∈ M2,2(R) and define hA, Bi = ae + bf + 2cg + 2dh. (a) Show hA, Bi is an inner product of M2,2(R).
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Solution 1

To show that hA, Bi is an inner product on M2,2(R), we need to verify the four properties of an inner product:

  1. Conjugate Symmetry: hA, Bi = hB, Ai
  2. Linearity in the first argument: hλA + μB, Ci = λhA, Ci + μhB, Ci for all λ, μ in R and all A, B, C in M2,2(R)
  3. Positive-definiteness: hA, Ai ≥ Knowee AI StudyGPT is a powerful AI-powered study tool designed to help you to solve study prob
Knowee AI StudyGPT is a powerful AI-powered study tool designed to help you to solve study problem.
Knowee AI StudyGPT is a powerful AI-powered study tool designed to help you to solve study problem.
Knowee AI StudyGPT is a powerful AI-powered study tool designed to help you to solve study problem.
Knowee AI StudyGPT is a powerful AI-powered study tool designed to help you to solv

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