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Linear Algebra With Applications
Found in: Page 289
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

Find the determinants of the linear transformations in Exercises 17 through 28.

21. T(f(t))=f(-t) from P3 to P3

Therefore, the determinant of the linear transformations is given by,

det T = det B = 1

See the step by step solution

Step by Step Solution

Step 1: Definition. 

A determinant is a unique number associated with a square matrix.

A determinant is a scalar value that is a function of the entries of a square matrix.

It is the signed factor by which areas are scaled by this matrix. If the sign is negative the matrix reverses orientation.

Step 2: Given. 

Given linear transformation,

T(f(t))=f(-t) from P3 to P3

Step 3: To find determinant.

Consider the linear transformation T:P3P3defined as T(f)=f(-t).

Consider the basis B=1,t,t2,t3 for P3.

We have that

T1=1T1=11+0t+0t2+0t3T(t)=-tT(t)=0(1)+(-1)t+0t2+0t3T(t2)=(-t)2 T(t2)=t2=0(1)+0t+1t2+0t3 andT(t3)=(-t)3 T(t3)=-t3T(t3)=0(1)+0t+0t2+(-1)t3

Thus the matrix of T with respect tois


Observe that



det T = det B =1

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