Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

Find the basis of all $P_{n}$ , and determine its dimension.

The dimension of $P_{n}$ is $n + 1$ which is spanned by role="math" localid="1659419285127" $Span \{1, t, t^{2}, . . ., t^{n}\}$ .

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Determine the span

Consider the set of all polynomial $P_{n}$ .

The set role="math" localid="1659419586773" ${1, t, t_{2}, . . ., t_{n}}$ is linear independent set of $V$ if there exist constant $a_{i} \in R$ such that $a_{0} + a_{1} t_{1} + a_{2} t_{2} + . . . + a_{n} t_{n} = 0$ where $a_{0} = a_{1} = a_{2} = . . . = a_{n} = 0$ .

Any polynomial $f (t) \in P_{n}$ is defined as follows.

$f (t) = b_{0} + b_{1} t + b_{2} t^{2} + . . . + b_{n} t^{n}$

Step 2: Compare the polynomials

Compare the equations $b_{0} + b_{1} t + b_{2} t^{2} + . . . + b_{n} t^{n} = 0$ bot side as follows.

$\begin{array}{l} b_{0} + b_{1} t + b_{2} t^{2} + . . . + . + b_{n} t^{n} = 0 \\ b_{0} + b_{1} t + b_{2} t^{2} + . . . + . + b_{n} t^{n} = (0) + (0) t + (0) t^{2} + . . . + (0) t^{n} \\ b_{i} = 0 \end{array}$

By the definition of linear independence, the subset $\{1, t, t^{2}, . . ., t^{n}\}$ is L.I where are linear independent.

Therefore, the set of polynomial $P_{n}$ is spanned by $S p a n \{1, t, t^{2}, . . ., t^{n}\}$ .

Hence, the dimension of $A$ is and $n + 1$ spanned by $S p a n 1, t, t^{2}, . . ., t^{n} .$

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Linear Algebra With Applications

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Short Answer

Find the basis of all $P_{n}$ , and determine its dimension.

Step by Step Solution

Step 1: Determine the span

Step 2: Compare the polynomials

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Linear Algebra With Applications

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Short Answer

Find the basis of all Pn, and determine its dimension.

Step by Step Solution

Step 1: Determine the span

Step 2: Compare the polynomials

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Find the basis of all $P_{n}$ , and determine its dimension.