Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

Find a basis of a linear space and thus determine its dimension. Examine whether a subset of a linear space is a subspace. Which of the subsets of $P_{2}$ given in Exercises 1 through 5 are subspaces of $P_{2}$ (see Example 16)? Find a basis for those that are subspace,
${p (t) : \int_{0}^{1} p (t) dt = 0}$ .

${p (t) : \int_{0}^{1} p (t) dt = 0}$ is a subset and a subspace of $P_{2}$ .

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Definition of subspace

A subset W of a linear space V is called a subspace of V if

(a) W contains the neutral element 0 of V.

(b) W is closed under addition (if f and g are in W then so is f+g )

(c) W is closed under scalar multiplication (if f is in W and k is scalar, then kf is in W ).

we can summarize parts b and c by saying that W is closed under linear combinations.

Step 2: Application of given conditions on vector space V.

General expression for a quadratic polynomial with one variable is $p (t) = {at}^{2} + bt + c$

Given $\int_{0}^{1} p (t) dt = 0$ .

From given,

$\int_{0}^{1} p (t) dt = \int_{0}^{1} {at}^{2} + bt + c$ .

Then,

$\int_{0}^{1} {at}^{2} + bt + c = {[\frac{{at}^{3}}{3} + \frac{{bt}^{2}}{2} + ct]}_{0}^{1} = \frac{a}{3} + \frac{b}{2} + c c = - \frac{a}{3} + \frac{b}{2} Therefore, p (t) = {at}^{2} + bt - \frac{a}{3} + \frac{b}{2} .$

Assume, $q (t) = {et}^{2} + dt - \frac{e}{3} - \frac{d}{2} .$ .

Also, let us assume two scalars $α, β$ ,

Then, we can write as:

$α p (t) + β q (t) = α (a t^{2} + b t - \frac{a}{3} - \frac{b}{2}) + β (e t^{2} + d t - \frac{e}{3} - \frac{d}{2}) = α a t^{2} + α b t - \frac{a}{3} α - \frac{b}{2} α + β e t^{2} + β d t - \frac{e}{3} β - \frac{d}{2} β = t^{2} (α a + β e) + t (α b + β d) - \frac{1}{3} (α a + β e) - \frac{1}{2} (α b + β d)$

Therefore, the given set is closed under linear combinations. Hence, it is a subspace of $P_{2}$ .

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Linear Algebra With Applications

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Short Answer

Step by Step Solution

Step 1: Definition of subspace

Step 2: Application of given conditions on vector space V.

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Linear Algebra With Applications

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Short Answer

Find a basis of a linear space and thus determine its dimension. Examine whether a subset of a linear space is a subspace. Which of the subsets of P2 given in Exercises 1 through 5 are subspaces of P2(see Example 16)? Find a basis for those that are subspace,{p(t):∫01p(t)dt=0}.

Step by Step Solution

Step 1: Definition of subspace

Step 2: Application of given conditions on vector space V.

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