Q70E Let聽聽a1,...,anbe distinct real... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 4: Q70E (page 198)

Let $a_{1}, . . ., a_{n}$ be distinct real numbers.Show that there exist 鈥渨eights鈥� $w_{1}, . . ., w_{n}$ such that
${鈭�}_{- 1}^{1} f (t) d t = {鈭�}_{i = 1}^{n} w_{i} f (a_{i})$
For all polynomials f(t) in $P_{n - 1}$

Short Answer

Expert verified

The solution is ${鈭�}_{- 1}^{1} f (t) d t = {鈭�}_{i = 1}^{n} w_{i} f (a_{i})$

Step by step solution

Step 1:Definition of isomorphism

An invertible linear transformation T is called an isomorphism.

Also says that the linear space V is isomorphic to the linear space W if there exists an isomorphism T from V to W.

Step:2 Determination of the invertible matrix

Consider a basis $f_{1}, . . ., f_{n}$ of $P_{n - 1}$ .

Let $a_{1}, . . ., a_{n}$ be distinct real numbers.

Consider the $n 脳 n$ matrix M whose ijth entry is $f_{j} (a_{i})$ .

Assume that the basis $B = (f_{1}, . . ., f_{n})$ is the basis of a linear space V then the coordinate transformation $L_{B} (f) = {[f]}_{B}$ from V to Rⁿ is an isomorphism

Now to show that there exist 鈥渨eights鈥� $w_{1}, . . ., w_{n}$ such that ${鈭�}_{- 1}^{1} f (t) d t = {鈭�}_{i = 1}^{n} w_{i} f (a_{i})$ .

Here by the properties of the isomorphisms we have that the weights $w_{1}, . . ., w_{n}$ obtains the function ${鈭�}_{- 1}^{1} f (t) d t = {鈭�}_{i = 1}^{n} w_{i} f (a_{i})$ for all the polynomials in $P_{n - 1}$

Hence the proof.

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91影视

Let $a_{1}, . . ., a_{n}$ be distinct real numbers.Show that there exist 鈥渨eights鈥� $w_{1}, . . ., w_{n}$ such that
${鈭�}_{- 1}^{1} f (t) d t = {鈭�}_{i = 1}^{n} w_{i} f (a_{i})$
For all polynomials f(t) in $P_{n - 1}$

Short Answer

Step by step solution

Step 1:Definition of isomorphism

Step:2 Determination of the invertible matrix

One App. One Place for Learning.

Most popular questions from this chapter

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91影视

Leta1,...,anbe distinct real numbers.Show that there exist 鈥渨eights鈥�w1,...,wnsuch that鈭�-11f(t)dt=鈭�i=1nwif(ai)For all polynomials f(t) inPn-1

Short Answer

Step by step solution

Step 1:Definition of isomorphism

Step:2 Determination of the invertible matrix

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Geometry

Mechanics Maths

Statistics

Probability and Statistics

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

Let $a_{1}, . . ., a_{n}$ be distinct real numbers.Show that there exist 鈥渨eights鈥� $w_{1}, . . ., w_{n}$ such that
${鈭�}_{- 1}^{1} f (t) d t = {鈭�}_{i = 1}^{n} w_{i} f (a_{i})$
For all polynomials f(t) in $P_{n - 1}$