Suppose A is an m × n matrix and B is an n × p matrix. Show thatCol AB ⊆ Col A and Nul B ⊆ Nul AB.Hence, rank (AB) ≤ rank A and rank (AB) ≤ rank B.

Question

Suppose A is an m × n matrix and B is an n × p matrix. Show thatCol AB ⊆ Col A and Nul B ⊆ Nul AB.Hence, rank (AB) ≤ rank A and rank (AB) ≤ rank B.
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Solution 1

Sure, let's break this down:

  1. Col AB ⊆ Col A: Let's take any vector v in Col AB. By definition of matrix multiplication, v can be written as ABu for some vector u in R^n. Now, ABu is a linear combination of the columns of A, with the entries of Bu as weights. Therefore, v is in the column Knowee AI StudyGPT is a powerful AI-powered study tool designed to help you to solve study prob
Knowee AI StudyGPT is a powerful AI-powered study tool designed to help you to solve study problem.
Knowee AI StudyGPT is a powerful AI-powered study tool designed to help you to solve study problem.
Knowee AI StudyGPT is a powerful AI-powered study tool designed to help you to solve study problem.
Knowee AI StudyGPT is a powerful AI-powered study tool designed to help you to solv

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