Q52E If A and B are n 脳 m matrices s... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 3: Q52E (page 165)

If A and B are n 脳 m matrices such that the image of A is a subset of the image of B, then there must exist an m 脳 m matrix C such that A = BC.

Short Answer

Expert verified

The above statement is true.

If A and B are n 脳 m matrices such that the image of A is a subset of the image of B, then there must exist an m 脳 m matrix C such that A = BC.

Step by step solution

Definition of the image of a matrix

Let A be a matrix from ${鈩�}^{n} to {鈩�}^{m}$ then the image of the matrix A, denoted by Im(A), is defined as-

$Im (A) = \{A x : x 鈭� {鈩�}^{n}\}$

To prove A = BC.

Let us consider two n x m matrices A and B such that

$A = [{\overset{鈫�}{u}}_{1} {\overset{鈫�}{u}}_{2} . . . {\overset{鈫�}{u}}_{m}]$ and $B = [{\overset{鈫�}{v}}_{1} {\overset{鈫�}{v}}_{2} . . . {\overset{鈫�}{v}}_{m}]$ .

Since, the image of matrix A is a subset of matrix B, then the vectors ${\overset{鈫�}{u}}_{1}, {\overset{鈫�}{u}}_{2}, . . ., {\overset{鈫�}{u}}_{m}$ are in the image of matrix B. So we can write

${\overset{鈫�}{u}}_{1} = B {\overset{鈫�}{w}}_{1}, {\overset{鈫�}{u}}_{2} = B {\overset{鈫�}{w}}_{2}, . . ., {\overset{鈫�}{u}}_{m} = B {\overset{鈫�}{w}}_{m}$

for some vectors ${\overset{鈫�}{w}}_{1}, {\overset{鈫�}{w}}_{2}, . . ., {\overset{鈫�}{w}}_{m} 鈭� {鈩�}^{m}$ .

Therefore $A = [{\overset{鈫�}{u}}_{1} {\overset{鈫�}{u}}_{2} . . . {\overset{鈫�}{u}}_{m}] = B [{\overset{鈫�}{w}}_{1} {\overset{鈫�}{w}}_{2} . . . {\overset{鈫�}{w}}_{m}] = B C$

$鈬� A = B C$

Where $C = [{\overset{鈫�}{w}}_{1} {\overset{鈫�}{w}}_{2} . . . {\overset{鈫�}{w}}_{m}]$ is an m x m matrix.

Final Answer

If A and B are n 脳 m matrices such that the image of A is a subset of the image of B, then there must exist an m 脳 m matrix C such that A = BC.

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91影视

If A and B are n 脳 m matrices such that the image of A is a subset of the image of B, then there must exist an m 脳 m matrix C such that A = BC.

Short Answer

Step by step solution

Definition of the image of a matrix

To prove A = BC.

Final Answer

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