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Solution: Exercises 12–17 develop properties of rank that
Chapter , Problem 15E(choose chapter or problem)
Exercises 12–17 develop properties of rank that are sometimes needed in applications. Assume the matrix A is m × n.Let A be an m × n matrix, and let B be an n × p matrix such that AB = 0. Show that rank A + rank B n. [Hint: One of the four subspaces Nul A, Col A, Nul B, and Col B is contained in one of the other three subspaces.]
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QUESTION:
Exercises 12–17 develop properties of rank that are sometimes needed in applications. Assume the matrix A is m × n.Let A be an m × n matrix, and let B be an n × p matrix such that AB = 0. Show that rank A + rank B n. [Hint: One of the four subspaces Nul A, Col A, Nul B, and Col B is contained in one of the other three subspaces.]
ANSWER:Solution 15EStep 1 of 2Consider A be an matrix and B be an matrix such that The objective is to show that .Let the columns of B be ,Then , implies, This is true for So, all the elements of column space of B must be contained in the null space of A.Since null space of A is a subspace of , all linear combination of the columns of B are in null space of A.By the definition of dimension,