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Linear Algebra With Applications
Found in: Page 185
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

T denotes the space of infinity sequence of real numbers, T(f(t))=[f7f11] from P2 to R2 .

The function T is linear but not isomorphism.

See the step by step solution

Step by Step Solution

Step 1: Determine the linearity of T.

[f(7)f(11)] for T(f(t)) and [g(7)g(11)] for T(g(t)) in T(f(t))+T(g(t)) as followsConsider the function T(f(t))=[f(7)f(11)] from P2 to R2

A function is called a linear transformation on if the function satisfies the following properties.

  1. D(x+y)=D(x)+D(y) for all x,yR.
  2. D(αx)=αD(x) for all constant αR.

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 localid="1659410729239" T(f(t))=[f(7)f(11)] and T(g(t))=[g(7)g(11)]and .

Substitute the value [f(7)f(11)] for T(f(t)) and [g(7)g(11)] for T(g(t)) in T(f(t))+T(g(t)) as follows

T(f(t))+T(g(t))=[f(7)f(11)]+ [g(7)g(11)]

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

localid="1659410186169" T(f+g(t))=f+g7f+g11 =f7 +g7f11+g11 =f7f11+g7g11T(f+g(t))=Tft+Tgt'

Assume localid="1659410277772" fP2 and αR then T(f(t))=[f(7)f(11)]and then .

Substitute the value [αf(7)αf(11)] for T(f(t)) as follows.

T(αf(t))=[αf(7)αf(11)] =α[f(7)f(11)] T(αf(t))=αT(f(t))

As T(f+g(t)) =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 isomorphism of  T.

As the function T define from P2 to R2 and P2 is spanned by 1,t,t2 means dimension of P2 is 3 and dimension of R2 is 2.

By the definition of isomorphism, the function T is not isomorphism.

Hence, the transformation T(f(t))=[f(7)f(11)] is linear but not isomorphism.

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