Chapter 5.5 Problem #32 from Linear Algebra 4th Edition by Otto Bretscher. In th

Question

Chapter 5.5 Problem #32 from Linear Algebra 4th Edition by Otto Bretscher. In the space C[-1, 1], we introduce the inner product = 1/2?1?1f(t)g(t)dt. (a) Find (t^m, t^n), where n and m are positive integers.(b) Find the norm of f(t) = t^n, where n is a positive integer.(c) Applying the Gram-Schmidt process to the standard basis, 1, t, t^2, t^3 of P3, construct an orthonormal basis g0(t), ..., g3(t) of P3 for the given inner product.(d) Find the polynomials (g0(t)) / (g0(1)), ..., (g3(t)) / (g3(1)). (Those are the first few Legendre Polynomials, named after the great French mathematician Adrien-Marie Legendre, 1752-1833. These polynomials have a wide range of applications in math, physics, and engineering. Note that the Legendre polynomials are normalized so that their value at 1 is 1.)(e) Find the polynomial g(t) in P3 that best approximates the function f(t) = (1) / (1+t^2) on the interval [-1, 1], for the inner product introduced in this exercise. Draw a sketch.

Explanation / Answer

Main article: Matrix multiplication Schematic depiction of the matrix product AB of two matrices A and B. Multiplication of two matrices is defined only if the number of columns of the left matrix is the same as the number of rows of the right matrix. If A is an m-by-n matrix and B is an n-by-p matrix, then their matrix product AB is the m-by-p matrix whose entries are given by dot product of the corresponding row of A and the corresponding column of B: , where 1 = i = m and 1 = j = p.[6] For example, the underlined entry 2340 in the product is calculated as (2 × 1000) + (3 × 100) + (4 × 10) = 2340: Matrix multiplication satisfies the rules (AB)C = A(BC) (associativity), and (A+B)C = AC+BC as well as C(A+B) = CA+CB (left and right distributivity), whenever the size of the matrices is such that the various products are defined.[7] The product AB may be defined without BA being defined, namely if A and B are m-by-n and n-by-k matrices, respectively, and m ? k. Even if both products are defined, they need not be equal, i.e., generally one has AB ? BA, i.e., matrix multiplication is not commutative, in marked contrast to (rational, real, or complex) numbers whose product is independent of the order of the factors. An example of two matrices not commuting with each other is: whereas The identity matrix In of size n is the n-by-n matrix in which all the elements on the main diagonal are equal to 1 and all other elements are equal to 0, e.g. It is called identity matrix because multiplication with it leaves a matrix unchanged: MIn = ImM = M for any m-by-n matrix M. Besides the ordinary matrix multiplication just described, there exist other less frequently used operations on matrices that can be considered forms of multiplication, such as the Hadamard product and the Kronecker product.[8] They arise in solving matrix equations such as the Sylvester equation.

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