Q23E In the space of the polynomials ... [FREE SOLUTION]

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Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 5: Q23E (page 261)

In the space of the polynomials of degree, we define the inner product
$< f, g > = \frac{1}{2} (f (0) + g (0) + f (1) + g (1))$
Find an orthonormal basis for this inner product space.

Short Answer

Expert verified

The orthonormal basis of $P_{1}$ is $\{1, 2 t - 1\}$ .

Step by step solution

Determine the Gram-Schmidt process.

Consider a basis of a subspace Vof $R^{n}$ for j = 2,...,m we resolve the vector ${\overset{鈫�}{v}}_{j}$ into its components parallel and perpendicular to the span of the preceding vectors localid="1660108333006" ${\overset{鈫�}{v}}_{1, . . .,} {\overset{鈫�}{v}}_{j - 1}$ ,

Then ${\overset{鈫�}{u}}_{1} = \frac{1}{|| {\overset{鈫�}{v}}_{1} ||} {\overset{鈫�}{v}}_{1}, {\overset{鈫�}{u}}_{2} = \frac{1}{|| {\overset{鈫�}{v}}_{2}^{鈯�} ||} {\overset{鈫�}{v}}_{2}^{鈯�}, . . ., {\overset{鈫�}{u}}_{j} = \frac{1}{|| {\overset{鈫�}{v}}_{j}^{鈯�} ||} {\overset{鈫�}{v}}_{j}^{鈯�}, . . ., {\overset{鈫�}{u}}_{m} = \frac{1}{|| {\overset{鈫�}{v}}_{m}^{鈯�} ||} {\overset{鈫�}{v}}_{m}$

It will use the Gram-Schmidt process to get an orthonormal basis.

Let $f_{1} (t) = 1, f_{2} (t) = t$ for find the first element of orthonormal basis is,

${||f||}^{2} = \frac{1}{2} (f (0) + f (0) + f (1) + f (1)) = 1$

Therefore take

Now, observe the following

$g_{2} (t) = \frac{f_{2} (t) - (g_{1} (t), f_{2} (t) g_{1} (t))}{||f_{2} (t) - (g_{1} (t) . f_{2} (t) g_{1} (t))||} = \frac{t^{2} - <1, t> 1}{||t^{2} - <1, t> 1||} = \frac{t - 1 / 2}{t - 1 / 2} = \frac{t - 1 / 2}{\sqrt{\frac{1}{2} (0 - \frac{1}{2} . 0 - \frac{1}{2} . + 1 - \frac{1}{2} . + 1 - \frac{1}{2})}}$

Further solve the above expression,

$g_{2} (t) = \frac{t - 1 / 2}{1 / 2} = (2 t - 1)$

Hence, an orthonormal basis of $P_{1}$ is $\{1, 2 t - 1\}$

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91影视

In the space of the polynomials of degree, we define the inner product
$< f, g > = \frac{1}{2} (f (0) + g (0) + f (1) + g (1))$
Find an orthonormal basis for this inner product space.

Short Answer

Step by step solution

Determine the Gram-Schmidt process.

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91影视

In the space of the polynomials of degree, we define the inner product<f,g>=12(f0+g0+f1+g1)Find an orthonormal basis for this inner product space.

Short Answer

Step by step solution

Determine the Gram-Schmidt process.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Discrete Mathematics

Pure Maths

Mechanics Maths

Geometry

Calculus

Study anywhere. Anytime. Across all devices.

In the space of the polynomials of degree, we define the inner product
$< f, g > = \frac{1}{2} (f (0) + g (0) + f (1) + g (1))$
Find an orthonormal basis for this inner product space.