Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

Find the set of all polynomial $f (t)$ in $P_{2}$ such that $f (1) = 0$ , and determine its dimension.

The dimension of $P_{2}$ such that $f (1) = 0$ is which is spanned by $S p a n {(t - 1), (t^{2} - 1)}$ .

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Determine the span set.

Consider the set of all polynomial such that .

The set ${1, t, t_{2}, . . ., t_{n}}$ is linear independent set V of 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 $f (t) \in P_{2}$ is defined as follows.

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

Substitute the value 1 for t in the equation $f (t) = b_{0} + b_{1} t + b_{2} t^{2}$ as follows.

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

As $f (1) = 0$ , substitute the value 0 for $f (1)$ in the equation $f (1) = b_{0} + b_{1} + b_{2}$ as follows.

$\begin{array}{l} f (1) = b_{0} + b_{1} + b_{2} \\ b_{0} = - (b_{1} + b_{2}) \end{array}$

Substitute the value $- (b_{1} + b_{2})$ for in the equation $f (t) = b_{0} + b_{1} t + b_{2} t^{2}$ as follows.

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

From the equation, the set $\{(t - 1), (t^{2} - 1)\}$ span $P_{2}$ if $f (1) = 0$ .

Step 2: Determine the property of independency.

Compare the equations $b_{1} (t - 1) + b_{2} (t^{2} - 1) = 0$ both side as follows.

$b_{1} (t - 1) + b_{2} (t^{2} - 1) = 0 b_{1} (t - 1) + b_{2} (t^{2} - 1) = 0 (t - 1) + (0) (t^{2} - 1) b_{i} = 0$

By the definition of linear independence, the subset $\{(t - 1), (t^{2} - 1)\}$ is L.I.

Therefore, the set of all polynomial $P^{2}$ such that $f (1) = 0$ has dimension 2 and spanned by $S p a n \{(t - 1), (t^{2} - 1)\}$ .

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Linear Algebra With Applications

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Short Answer

Find the set of all polynomial $f (t)$ in $P_{2}$ such that $f (1) = 0$ , and determine its dimension.

Step by Step Solution

Step 1: Determine the span set.

Step 2: Determine the property of independency.

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Linear Algebra With Applications

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Short Answer

Find the set of all polynomial f(t) in P2 such that f(1)=0, and determine its dimension.

Step by Step Solution

Step 1: Determine the span set.

Step 2: Determine the property of independency.

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Find the set of all polynomial $f (t)$ in $P_{2}$ such that $f (1) = 0$ , and determine its dimension.