Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

(a) Show that T is a linear transformation.
(b) Find the kernel of T.
(c) Show that the image of T is a space $L (R^{m}, R^{n})$ of all linear transformation $R^{m}$ to role="math" localid="1659420398933" $R^{n}$ .
(d) Find the dimension of $L (R^{m}, R^{n})$ .

(a) The solution is a linear transformation.

(b) The solution is the kernel of T contains only zero matrix.

(c) The solution is the image of T is the space $L (R^{m}, R^{n})$ of all linear transformation from $R^{m}$ to role="math" localid="1659420484059" $R^{n}$ .

(d) The solution is the dimension of $L (R^{m}, R^{n})$ is mn.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

(a) Step1: Definition of Linear Transformation

Consider two linear spaces V and W. A function T is said to be linear transformation if the following holds.

$T (f + g) = T (f) + T (g) T (kf) = kT (f)$

For all elements f,g of V and k is scalar.

Step2: Explanation of the solution

Consider the transformation as follows.

$T : R^{n \times m} \to F (R^{n}, R^{m})$ is defined by $(T (A)) (\vec{v}) = A \vec{v}$ .

Simplify for the linear transformation first condition as follows.

$(T (A)) (\vec{v} + \vec{w}) = A (\vec{v} + \vec{w}) = A \vec{v} + A \vec{w} (T (A)) (\vec{v} + \vec{w}) = (T (A)) \vec{v} + (T (A)) \vec{w}$

Similarly, simplify for the second condition as follows.

$(T (A)) (k \vec{v}) = A (k \vec{v}) = k (A \vec{v}) = K (T (A)) (\vec{v}) (T (A)) (k \vec{v}) = k (T (A)) (\vec{v})$

Thus, T is a linear transformation.

(b) Step3: Definition of image and kernel

A linear transformation $T : V \to W$ is said to be an isomorphism if and only if $\ker (T) = {0}$ and $im (T) = W$ or $\dim (V) = \dim (W)$ .

Step4: Explanation of the solution

Consider for a non-zero vector $\vec{v}$ as follows.

$(T (A)) (\vec{v}) = \vec{0} A \vec{v} = 0$

Then by equality of two matrix as follows.

$\vec{v} = \vec{0}$

Thus, the kernel of T contains only zero matrix.

(c) Step5: Definition of image and kernel

A linear transformation $T : V \to W$ is said to be an isomorphism if and only if $\ker (T) = {0}$ and $im (T) = W$ or $\dim (V) = \dim (W)$ .

Step6: Explanation of the solution

Since, the transformation is linear.

Therefore, the image of T is the linear space of all the linear transformation from $R^{m}$ to $R^{n}$ .

Hence, the image of T is the space $L (R^{m}, R^{n})$ of all linear transformation from $R^{m}$ to $R^{n}$ .

(d)Step7: Definition of Linear Transformation

Consider two linear spaces V and W. A function T is said to be linear transformation if the following holds.

$T (f + g) = T (f) + T (g) T = (kf) = kT (f)$

For all elements $f, g$ of V and k is scalar.

A linear transformation $T : V \to W$ is said to be an isomorphism if and only if $\ker (T) = {0}$ and $im (T) = W$ or $\dim (V) = \dim (W)$ .

Step8: Explanation of the solution

Consider the transformation as follows.

$T : R^{n \times m} \to F (R^{n}, R^{m})$ is defined by $(T (A)) (\vec{v}) = A \vec{v}$ .

The dimension of the space is as follows.

$\dim (L (R^{m}, R^{n})) = \dim (R^{n \times m}) = mn \dim (L (R^{m}, R^{n})) = mn$

Thus, the dimension of $L (R^{m}, R^{n})$ is mn.

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Linear Algebra With Applications

Answers without the blur.

Short Answer

(a) Show that T is a linear transformation.
(b) Find the kernel of T.
(c) Show that the image of T is a space $L (R^{m}, R^{n})$ of all linear transformation $R^{m}$ to role="math" localid="1659420398933" $R^{n}$ .
(d) Find the dimension of $L (R^{m}, R^{n})$ .

Step by Step Solution

(a) Step1: Definition of Linear Transformation

Step2: Explanation of the solution

(b) Step3: Definition of image and kernel

Step4: Explanation of the solution

(c) Step5: Definition of image and kernel

Step6: Explanation of the solution

(d)Step7: Definition of Linear Transformation

Step8: Explanation of the solution

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Linear Algebra With Applications

Answers without the blur.

Short Answer

(a) Show that T is a linear transformation.(b) Find the kernel of T.(c) Show that the image of T is a space L(Rm,Rn)of all linear transformation Rm to role="math" localid="1659420398933" Rn.(d) Find the dimension of L(Rm,Rn).

Step by Step Solution

(a) Step1: Definition of Linear Transformation

Step2: Explanation of the solution

(b) Step3: Definition of image and kernel

Step4: Explanation of the solution

(c) Step5: Definition of image and kernel

Step6: Explanation of the solution

(d)Step7: Definition of Linear Transformation

Step8: Explanation of the solution

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(a) Show that T is a linear transformation.
(b) Find the kernel of T.
(c) Show that the image of T is a space $L (R^{m}, R^{n})$ of all linear transformation $R^{m}$ to role="math" localid="1659420398933" $R^{n}$ .
(d) Find the dimension of $L (R^{m}, R^{n})$ .