Exercises 12–17 develop properties of rank that are

Chapter , Problem 12E

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QUESTION:

Exercises 12–17 develop properties of rank that are sometimes needed in applications. Assume the matrix A is \(m \times n\).

Show from parts (a) and (b) that rank AB cannot exceed the rank of A or the rank of B. (In general, the rank of a product of matrices cannot exceed the rank of any factor in the product.)

a. Show that if B is \(n \times p\), then rank \(A B \leq \operatorname{rank} A\). [Hint: Explain why every vector in the column space of AB is in the column space of A.]

b. Show that if B is \(n \times p\), then rank \(A B \leq \operatorname{rank} B\). [Hint: Use part (a) to study rank \((A B)^{T}\).]

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QUESTION:

Exercises 12–17 develop properties of rank that are sometimes needed in applications. Assume the matrix A is \(m \times n\).

Show from parts (a) and (b) that rank AB cannot exceed the rank of A or the rank of B. (In general, the rank of a product of matrices cannot exceed the rank of any factor in the product.)

a. Show that if B is \(n \times p\), then rank \(A B \leq \operatorname{rank} A\). [Hint: Explain why every vector in the column space of AB is in the column space of A.]

b. Show that if B is \(n \times p\), then rank \(A B \leq \operatorname{rank} B\). [Hint: Use part (a) to study rank \((A B)^{T}\).]

ANSWER:

Solution 12EStep 1 of 2(a)The objective is to show that if B is , then To show this, we have to explain why every vector in the column space of is in thecolumn space of AFor this, let be in column space of , then can be written as, , for some vector xThen, Since is also a vector, we conclude that is also in column space of A.So, column space of is a subset of column space of A.Since these sets are vector spaces, we conclude that

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