Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

Answers without the blur.

Just sign up for free and you're in.

Short Answer

Find the image, kernel, rank, and nullity of the transformation T in $T (f (t)) = t (f' (t)) from P_{2} to P_{2} .$

The image contains of all the linear polynomial of the form $2 a t^{2} + b t$ and the nullity is 1.

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step1: Definition of rank of T

Consider the transformation as follows.

$T : V \to W such that im (T) = {T (f) : f \in V}$

If the image of T is finite dimensional, then dim(imT) is called the rank of T .

Step2: Explanation of the solution

Consider the linear transformation as follows.

$T : P_{2} \to P_{2} d e f i n e d a s T (f (t)) = t (f' (t)) .$

$S i n c e, f (t) \in P_{2} .$

Consider the equation as follows.

$f (t) = a t^{2} + b t + c$

Simplify as follows.

$T (f (t)) = t (f' (t)) = t (\frac{d f}{d t} (a t^{2} + b t + c)) = t (2 a t + b) = 2 a t^{2} + b t$

Therefore, the image contains of all the second-degree polynomial of the form $2 a t^{2} + b t$ .

Thus, the rank is 2 with the basis $\{t, t^{2}\}$ .

Step3: Definition of nullity of T

Consider the transformation as follows.

$T : V \to W s u c h t h a t k e r (T) = {f \in V | T (f) = 0} .$

$I f t h e k e r n e l o f T i s f i n i t e d i m e n s i o n a l, t h e n d i m (k e r T) i s c a l l e d t h e n u l l i t y o f T$

Consider a constant polynomial as follows.

$T (f (t)) = 0$

Then, f(t) is a constant polynomial

Therefore, the kernel consists of all the constant polynomial with the basis {1} .

The nullity is 1.

Hence, the image contains of all the linear polynomial of the form $2 a t^{2} + b t$ and the rank is 2 whereas the kernel contains all the constant polynomial and the nullity is 1.

Find Study Materials for

Create Study Materials

Select your language

Linear Algebra With Applications

Answers without the blur.

Short Answer

Find the image, kernel, rank, and nullity of the transformation T in $T (f (t)) = t (f' (t)) from P_{2} to P_{2} .$

Step by Step Solution

Step1: Definition of rank of T

Step2: Explanation of the solution

Step3: Definition of nullity of T

Most popular questions for Math Textbooks

Want to see more solutions like these?

Recommended explanations on Math Textbooks

Select your language

Linear Algebra With Applications

Answers without the blur.

Short Answer

Find the image, kernel, rank, and nullity of the transformation T in T(f(t))=t(f'(t)) from P2 to P2.

Step by Step Solution

Step1: Definition of rank of T

Step2: Explanation of the solution

Step3: Definition of nullity of T

Most popular questions for Math Textbooks

Want to see more solutions like these?

Recommended explanations on Math Textbooks

Decision Maths

Calculus

Probability and Statistics

Statistics

Mechanics Maths

Geometry

Pure Maths

Find the image, kernel, rank, and nullity of the transformation T in $T (f (t)) = t (f' (t)) from P_{2} to P_{2} .$