Question

Let $A$ and $B$ be matrices of respective sizes $m \times n$ and $n \times p$. (a) Prove that $\operatorname{dim} \operatorname{ker}(A B) \leq \operatorname{dim} \operatorname{ker} A+\operatorname{dim} \operatorname{ker} B$. (b) Prove the Sylvester Inequalities $\operatorname{rank} A+\operatorname{rank} B-n \leq \operatorname{rank}(A B) \leq \min \{\operatorname{rank} A, \operatorname{rank} B\}$.

    Let $A$ and $B$ be matrices of respective sizes $m \times n$ and $n \times p$.
(a) Prove that $\operatorname{dim} \operatorname{ker}(A B) \leq \operatorname{dim} \operatorname{ker} A+\operatorname{dim} \operatorname{ker} B$.
(b) Prove the Sylvester Inequalities $\operatorname{rank} A+\operatorname{rank} B-n \leq \operatorname{rank}(A B) \leq \min \{\operatorname{rank} A, \operatorname{rank} B\}$.

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Applied Linear Algebra (Undergraduate Texts in Mathematics)

Applied Linear Algebra (Undergraduate Texts in Mathematics)

Peter J. Olver,… 2nd Edition

Chapter 2, Problem 42

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Let $A$ and $B$ be matrices of respective sizes $m \times n$ and $n \times p$. (a) Prove that $\operatorname{dim} \operatorname{ker}(A B) \leq \operatorname{dim} \operatorname{ker} A+\operatorname{dim} \operatorname{ker} B$. (b) Prove the Sylvester Inequalities $\operatorname{rank} A+\operatorname{rank} B-n \leq \operatorname{rank}(A B) \leq \min \{\operatorname{rank} A, \operatorname{rank} B\}$.

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