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\(\newcommand{\Cs}{\mathbb{Cs}} \newcommand{\Cm}{\mathbb{C}^{m}} \newcommand{\Cmm}{\mathbb{C}^{m\times m}} \newcommand{\Cmn}{\mathbb{C}^{m\times n}} \newcommand{\R}{\mathbb{R}} \newcommand{\Rm}{\mathbb{R}^{m}} \newcommand{\Rn}{\mathbb{R}^{n}} \newcommand{\Rmm}{\mathbb{R}^{m\times m}} \DeclareMathOperator{\Range}{range} \DeclareMathOperator{\Rank}{rank} \DeclareMathOperator{\diag}{diag} \DeclareMathOperator{\Span}{span} \DeclareMathOperator{\range}{range} \DeclareMathOperator{\Null}{null} \newcommand{\definiteintegral}[4]{\int_{#1}^{#2}\,#3\,d#4} \newcommand{\indefiniteintegral}[2]{\int#1\,d#2} \graphicspath{ {images/} } \usepackage{amsmath} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \)

Section3Range, nullspace, and rank (NM)

More from Lecture 1 of Trefethen and Bau.



Prove that \(A \in \Cmm\) has full rank if and only if it maps no two distinct vectors to the same vector. That is, show that if \(x\) and \(y\) are vectors, with \(x \neq y\text{,}\) then

\begin{equation*} Ax \neq Ay \iff \Rank(A)=m. \end{equation*}

Suppose you know the rank of the \(m\times m\) matrices \(A\) and \(B\text{.}\) What, if anything, can you say about the rank of \(C=AB\text{?}\) Suppose \(A\) or \(B\) have full rank, what can you say about the rank of \(C\text{?}\)