This exercise assumes familiarity with linear algebra. Let L: 𝐑^d →𝐑^d be a linear transformatio

This exercise assumes familiarity with linear algebra. Let...

This exercise assumes familiarity with linear algebra. Let $L: \mathbf{R}^d \rightarrow \mathbf{R}^d$ be a linear transformation. (1) Show that there exists a non-negative real number $D$ such that $m(L(E))=D m(E)$ for every elementary set $E$ (note from previous exercises that $L(E)$ is Jordan measurable). (Hint: apply Exercise 1.1.3 to the map $E \mapsto m(L(E))$.) (2) Show that if $E$ is Jordan measurable, then $L(E)$ is also, and $m(L(E))=D m(E)$. (3) Show that $D=|\operatorname{det} L|$. (Hint: Work first with the case when $L$ is an elementary transformation, using Gaussian elimination. Alternatively, work with the cases when $L$ is a diagonal transformation or an orthogonal transformation, using the unit ball in the latter case, and use the polar decomposition.)

This exercise assumes familiarity with linear algebra. Let $L: \mathbf{R}^d \rightarrow \mathbf{R}^d$ be a linear transformation.
(1) Show that there exists a non-negative real number $D$ such that $m(L(E))=D m(E)$ for every elementary set $E$ (note from previous exercises that $L(E)$ is Jordan measurable). (Hint: apply Exercise 1.1.3 to the map $E \mapsto m(L(E))$.)
(2) Show that if $E$ is Jordan measurable, then $L(E)$ is also, and $m(L(E))=D m(E)$.
(3) Show that $D=|\operatorname{det} L|$. (Hint: Work first with the case when $L$ is an elementary transformation, using Gaussian elimination. Alternatively, work with the cases when $L$ is a diagonal transformation or an orthogonal transformation, using the unit ball in the latter case, and use the polar decomposition.)

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