Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

State true or false, the linear transformation T (f) = f (4t - 3 ) from P to P is an isomorphism.

The statement is True

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Determine the linearity of T .

Consider the function T(f (t)) = f (4t - 3 ) from $P_{2} to P_{2}$ .

A function D is called a linear transformation on localid="1659425827307" $ℝ$ if the function D satisfies the following property’s.

$(a) D (x + y) = D (x) + D (y) for all x, y \in ℝ (b) D (αx) = αD (x) for all constant α \in ℝ$

An invertible linear transformation is called isomorphism or dimension of domain and co-domain is not same then the function is not isomorphism.

Assume $f, g \in P_{2} then T (f (t)) = f (4 t - 3) and T (g (t)) = g (4 t - 3)$ .

Substitute the value f ( 4t - 3 ) for T ( f (t) ) and g ( 4t - 3 ) for T ( f (t) ) in T ( f (t) ) +T ( f (t) ) as follows.

T ( f (t) ) +T ( f (t) ) =f (4t - 3) + g (4t - 3)

Now, simplify T ( { f + g } (t)) as follows.

T ( { f + g } (t)) = { f + g } ( 4t - 3)

T ( { f + g } (t)) = f ( 4t - 3) +g ( 4t - 3)

T ( { f + g } (t)) = T ( f(t) + T ( g (t))

$Assume f \in P_{2} and α \in ℝ then T ((α f)) = (α f) (4 t - 3) .$

Simplify the equation $T ((α f) (t)) = (α f) (4 t - 3) .$ as follows.

$T ((α f) (t)) = (α f) (4 t - 3) T ((α f) (t)) = α f (4 t - 3) T ((α f) (t)) = α T (f (t)) .$

As $T (\{f + g\} (t)) = T (f (t)) + T (g (t))$ and $T (α f (t)) = α T (f (t))$ , by the definition of linear transformation T is linear.

Step 2: Determine the rank of the transformation.

As $\{1, t, t^{2}, . . .\}$ is the basis element of P and the matrix corresponding to basis is upper triangular matrix.

Each row and column of upper triangular matrix is linearly independent.

Therefore, the rank (T) is n.

Step 3: Determine the dimension of kernel.

The Kernel (T) is defined as follows.

dim(T) = dim (Kernal (T)) + rank (T)

n = dim ( Kernel (T)) + 0

dim (Kernel (T)) = 0

Therefore, the dimension of is Kernel (T).

Step 4: Determine the isomorphism.

Theorem: Consider a linear transformation T defined from $T : V \to W$ then the transformation T is an isomorphism if and only if Ker (T) = 0 where .

$k e r (T) = \{f (x) \in P : T \{f (x)\} = 0\}$

As the dimension of Kernel (T) is 0, by the theorem the function T is an isomorphism.

Hence, the statement is True.

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Linear Algebra With Applications

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Short Answer

State true or false, the linear transformation T (f) = f (4t - 3 ) from P to P is an isomorphism.

Step by Step Solution

Step 1: Determine the linearity of T .

Step 2: Determine the rank of the transformation.

Step 3: Determine the dimension of kernel.

Step 4: Determine the isomorphism.

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Linear Algebra With Applications

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Short Answer

State true or false, the linear transformation T (f) = f (4t - 3 ) from P to P is an isomorphism.

Step by Step Solution

Step 1: Determine the linearity of T .

Step 2: Determine the rank of the transformation.

Step 3: Determine the dimension of kernel.

Step 4: Determine the isomorphism.

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