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Linear Algebra With Applications
Found in: Page 200
Linear Algebra With Applications

Linear Algebra With Applications

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

See the step by step solution

Step by Step Solution

Step 1: Determine the linearity of T .

Consider the function T(f (t)) = f (4t - 3 ) from P2 to P2.

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

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

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,gP2 then T ( f (t))=f(4t-3) and T (g(t))=g(4t-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 P2 and α then T((αf))=(αf)(4t-3).

Simplify the equation T((αf) (t))=(αf)(4t-3).as follows.

T((αf) (t))=(αf)(4t-3)T((αf)(t))=αf (4t-3)T((αf)(t))=αT(f(t)).

As T(f+gt)=T(f(t))+T(g(t))and T(αf(t))=αT(f(t)), by the definition of linear transformation T is linear.

Step 2: Determine the rank of the transformation.

As 1,t,t2,...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:VWthen the transformation T is an isomorphism if and only if Ker (T) = 0 where .

ker T=fx P:Tfx=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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