Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

Consider two n x m matrices A and B. What can you say about the relationship among the quantities rank(A), rank(B), rank(A+B).

If we have two n x m matrices A and B then

$rank (A + B) \leq rank (A) + rank (B) .$

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Mentioning concept

Since we know that

Dim (Im(A)) = rank(A) and dim (Im(B)) = rank(B)

So, dim (Im(A+B)) = rank(A+B)

Step 2: Finding the rank of (A+B)

Let $A = [{\vec{v}}_{1} {\vec{v}}_{2} . . . {\vec{v}}_{m}] and B = [w ⃗_{1} w ⃗_{2} . . . w ⃗_{m}]$

Where $v ⃗_{i} a n d w ⃗_{i}$ are column vectors of A and B respectively.

Then,

role="math" localid="1659355348377" $rank (A) = \dim (span (v ⃗_{1}, v ⃗_{2}, . . ., v ⃗_{m})) and rank (B) = \dim (span (w ⃗_{1}, w ⃗_{2}, . . . w ⃗_{m}))$

$\Rightarrow r a n k (A + B) = d i m (s p a n (v ⃗_{1} + w ⃗_{1}, v ⃗_{2} + w ⃗_{2}, . . . v ⃗_{m} + w ⃗_{m}))$

Step 3: Finding the dimension of image of (A+B)

Let us claim,

$s p a n (v ⃗_{1} + w ⃗_{1}, \vec{v} ⃗_{2} + w ⃗_{2}, . . ., v ⃗_{m} + w ⃗_{m}) \subset s p a n (v ⃗_{1}, v ⃗_{2}, . . ., v ⃗_{m}) + s p a n (w ⃗_{1}, w ⃗_{2}, . . ., w ⃗_{m})$

Then any vector can be written as

$x ⃗ = a_{1} (v ⃗_{1} + w ⃗_{1}) + a_{2} (v ⃗_{2} + w ⃗_{2}) + . . . + a_{m} (v ⃗_{m} + w ⃗_{m}) \Rightarrow x ⃗ = (a_{1} v ⃗_{1} + a_{2} v ⃗_{2} + . . . + a_{m} v ⃗_{m}) + (a_{1} w ⃗_{1} + a_{2} w ⃗_{2} + . . . + a_{m} w ⃗_{m}) \Rightarrow x ⃗ \in s p a n (v ⃗_{1}, v ⃗_{2}, . . ., v ⃗_{m}) + s p a n (w ⃗_{1}, w ⃗_{2}, . . ., w ⃗_{m})$

for some scalars $a ⃗_{1}, a ⃗_{2}, . . ., a_{m}$ .

$\Rightarrow s p a n (v ⃗_{1} + w ⃗_{1}, v ⃗_{2} + w ⃗_{2}, . . ., v ⃗_{m} + w ⃗_{m}) \subseteq s p a n (v ⃗_{1}, v ⃗_{2}, . . ., v ⃗_{m}) + s p a n (w ⃗_{1}, w ⃗_{2}, . . ., w ⃗_{m}) \Rightarrow I m (A + B) \subseteq I m (A) + I m (B) \Rightarrow d i m (I m (A + B)) \leq d i m (I m (A)) + d i m (I m (B)) \Rightarrow r a n k (A + B) \leq r a n k (A) + r a n k (B)$

Step 4: Final Answer

If we have two n x m matrices A and B then

$\Rightarrow I m (A + B) \subseteq I m (A) + I m (B) \Rightarrow d i m (I m (A + B)) \leq d i m (I m (A)) + d i m (I m (B)) \Rightarrow r a n k (A + B) \leq r a n k (A) + r a n k (B)$

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Linear Algebra With Applications

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Short Answer

Consider two n x m matrices A and B. What can you say about the relationship among the quantities rank(A), rank(B), rank(A+B).

Step by Step Solution

Step 1: Mentioning concept

Step 2: Finding the rank of (A+B)

Step 3: Finding the dimension of image of (A+B)

Step 4: Final Answer

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Linear Algebra With Applications

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Short Answer

Consider two n x m matrices A and B. What can you say about the relationship among the quantities rank(A), rank(B), rank(A+B).

Step by Step Solution

Step 1: Mentioning concept

Step 2: Finding the rank of (A+B)

Step 3: Finding the dimension of image of (A+B)

Step 4: Final Answer

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