please show the solution of (b), thanks ( )2. Let A denote the 2 x 2 matrixWe define a function (·, ·) : R² × R² → R by taking(x, y) = y" Ax,for each pair of vectors x, y E R?. (Technically, the result of computing yAx is a 1 x 1 matrix, not areal number. However, we identify the 1 x 1 matrix with its single real entry.)(a) Does (, ·) define an inner product on R²? If it does, prove it. If it does not, state which innerproduct conditions this function satisfies, and prove those. Also, provide counterexamples foreach inner product condition that this function does not satisfy.(b) By replacing A with another matrix B different from A in only a single entry, define an innerproduct [', ·] different from the function in part (a), and justify why this modified function is aninner product.(c) Give an orthonormal basis for R? with respect to this newly-defined inner product [', ] that youchose in part (b). Justify that your chosen basis is orthonormal.

To prove a function to be an inner product, we have to show it satisfies the following five…

(b) By replacing A with another matrix B different from A in only a single entry, define an inner product [, ] different from the function in part (a), and justify why this modified function is an inner product.

(b) By replacing A with another matrix B different from A in only a single entry, define an inner product [, ] different from the function in part (a), and justify why this modified function is an inner product.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter4: Eigenvalues And Eigenvectors

Section4.6: Applications And The Perron-frobenius Theorem

Problem 70EQ

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Question

please show the solution of (b), thanks

( )
2. Let A denote the 2 x 2 matrix
We define a function (·, ·) : R² × R² → R by taking
(x, y) = y" Ax,
for each pair of vectors x, y E R?. (Technically, the result of computing yAx is a 1 x 1 matrix, not a
real number. However, we identify the 1 x 1 matrix with its single real entry.)
(a) Does (, ·) define an inner product on R²? If it does, prove it. If it does not, state which inner
product conditions this function satisfies, and prove those. Also, provide counterexamples for
each inner product condition that this function does not satisfy.
(b) By replacing A with another matrix B different from A in only a single entry, define an inner
product [', ·] different from the function in part (a), and justify why this modified function is an
inner product.
(c) Give an orthonormal basis for R? with respect to this newly-defined inner product [', ] that you
chose in part (b). Justify that your chosen basis is orthonormal.

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