Problem 1E: In C[a,b] , define the product f,g=abf(t)g(t)dt . Show that this product satisfies the property f,g0... Problem 2E: Does the equation f,g+h=f,g+f,h hold for all elements f, g, h of an inner product space?Explain. Problem 3E: Consider a matrix S in nn . In n , define the product x,y=(Sx)TSy . a. For matrices S is this an... Problem 4E: In nm , consider the inner product A,B=trace(ATB) defined in Example 3. a. Find a formula for this... Problem 5E: Is A,B=trace(ABT) an inner product in nm ?(The notation A,B is chosen to distinguish this product... Problem 6E: a. Consider an nm matrix P and an mn matrixQ. Show that trace(PQ)=trace(QP) . b. Compare the... Problem 7E Problem 8E Problem 9E Problem 10E: Consider the space P2 with inner product f,g=1211f(t)g(t)dt .Find an orthonormal basis of the space... Problem 11E Problem 12E Problem 13E: For a function f in C[,] (with the inner productdefined on page 255), consider the sequence of all... Problem 14E: Which of the following is an inner product in P2 ?Explain. a. f,g=f(1)g(1)+f(2)g(2) b.... Problem 15E Problem 16E Problem 17E Problem 18E Problem 19E Problem 20E Problem 21E Problem 22E Problem 23E Problem 24E: Consider the linear space P of all polynomials, with inner product f,g=01f(t)g(t)dt . For three... Problem 25E Problem 26E Problem 27E: Find the Fourier coefficients of the piecewise continuous function f(t)={0ift01ift0. Problem 28E Problem 29E Problem 30E Problem 31E: Gaussian integration,In an introductory calculuscourse, you may have seen approximation formulas... Problem 32E: In the space C[1,1] , we introduce the inner product f,g=11f(t)g(t)dt . a. Find tn,tm , where n and... Problem 33E: a. Let w(t) be a positive-valued function in C[a,b] ,where ba . Verify that the rule... Problem 34E: In the space C[1,1] , we define the inner product f,g=1121 t 2f(t)g(t)dt=2111 t 2f(t)g(t)dt . See... Problem 35E: In this exercise, we compare the inner products andnorms introduced in Problems 32 and 34. Let’s... format_list_bulleted