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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))=[f(5)f(7)f(11)] from P2 to R3. .

The function T is linear and isomorphism.

See the step by step solution

Step by Step Solution

Step 1: Determine the linearity of T.

Consider the function T(f(t))=[f(5)f(7)f(11)] from P2 to R3.

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,gP2t then T(f(t))=[f(5)f(7)f(11)] and T(g(t))=[g(5)g(7)g(11)] .

Substitute the value role="math" localid="1659411768440" [f(5)f(7)f(11)] for T(f(t)) and [g(5)g(7)g(11)] for T(g(t)) in T(f(t))+T(g(t)) as follows.

T(f(t))+T(g(t))=[f(5)f(7)f(11)] + [g(5)g(7)g(11)]

Step 2: Simplify further

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

T(f+g(t))=f+g5f+g7f+g11 =f5g5f7g7f11g11 =f5f7f11+g5g7g11 T(f+g(t))= T(f(t)) +T(g(t))

Assume fP2 and αR then role="math" localid="1659412647658" T(f(t))=f5f7f11.

Substitute the value αf5αf7αf11 for T(αf(t))for as follows.

T(αf(t))=αf5αf7αf11 =αf5f7f11 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 3: Determine the isomorphism of .

As the function define from P2 to R3 and P2 is spanned by 1,t,t2 means dimension of P2 is 3 and dimension of R3 is 3.

By the definition of polynomial, every polynomial form P2 is described in a unique way means for every f5f7f11R3 there exist a unique f(t)P2 such that T(f(t))=f5f7f11 .

By the definition of isomorphism, the function is isomorphism.

Hence, the transformation localid="1662122176677" T(f(t))=f5f7 is linear and isomorphism.

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