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Linear Algebra With Applications
Found in: Page 146
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

Consider a 4 x 2 matrix A and 2 x 5 matrix B.

a. What are the possible dimensions of the kernel of AB?

b. What are the possible dimensions of the image of AB?

a. Possible dimensions of the kernel of AB = {5,4,3,2,1}

b. Possible dimensions of the image of AB ={0,1,2,3,4}.

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1:   Mentioning the concept

For any matrix A of order n x m

Dim (Im(A)) = rank(A).

And

Dim (ker(A))= number of free variables

=total number of variables −number of leading variables

= m − rank(A).

Step 2:   Given quantities

We have given two matrices A of order 4 x 2, and B of order 2 x5

Then the matrix AB is of order 4 x 5.

Now we have to find the possible dimensions of ker(AB) and Im(AB).

Step 3:   (b) Finding the possible dimensions of Im(AB).

Now, for a n x m matrix A, rank (A) ≤ min (m,n).

Then for a matrix AB of order 4 x 5, rank (AB) ≤ min (4,5)

⇒rank(AB)≤4⇒ran(AB)=0 or 1 or 2 or 3 or 4

Thus, possible value of rank(AB)={0,1,2,3,4}.

Since, dim(Im(AB))=rank(AB)

Then possible dimensions of image of AB = {0,1,2,3,4}.

Step 4:  (a) Finding the possible dimensions of ker(AB).

Since the number of unknown variables in an n x m matrix = m then the number of unknown variables in a 4 x 5 matrix = 5.

Now, the dim(ker(A)) = m – rank(AB) = 5 - {0,1,2,3,4}.

⇒ Possible values of dim(ker(AB)) = {5,4,3,2,1}

Step 5: Final Answer

If we have a 4 x 2 matrix A and a 2 x 5 matrix B then .

Then,

c. Possible dimensions of the kernel of AB = {5,4,3,2,1}

d. Possible dimensions of the image of AB ={0,1,2,3,4}.

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