2. Let u-("і "2,W3), ν (η , v 2, vs), and w-( w. w2,W3 ) be three vectors in R. Recall that the oriented volume V(u,v,w) of the parallelepiped spanned by u,v,w equals (их v) . w (See p. 45 for the definition of the cross product.) (b) If A is a 3 x 3-matrix, prove that V(Au,Av,Aw)-det(A)V(u, v, w). Remark. The formula from (b) shows that det(A) measures how the linear transforma tion with matrix A changes the volume of a region in R5.

Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
Chapter1: Line And Angle Relationships
Section1.4: Relationships: Perpendicular Lines
Problem 27E
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2. Let u-("і "2,W3), ν
(η , v 2, vs), and w-( w. w2,W3 ) be three vectors in R. Recall that
the oriented volume V(u,v,w) of the parallelepiped spanned by u,v,w equals (их v) . w
(See p. 45 for the definition of the cross product.)
(b) If A is a 3 x 3-matrix, prove that
V(Au,Av,Aw)-det(A)V(u, v, w).
Remark. The formula from (b) shows that det(A) measures how the linear transforma
tion with matrix A changes the volume of a region in R5.
Transcribed Image Text:2. Let u-("і "2,W3), ν (η , v 2, vs), and w-( w. w2,W3 ) be three vectors in R. Recall that the oriented volume V(u,v,w) of the parallelepiped spanned by u,v,w equals (их v) . w (See p. 45 for the definition of the cross product.) (b) If A is a 3 x 3-matrix, prove that V(Au,Av,Aw)-det(A)V(u, v, w). Remark. The formula from (b) shows that det(A) measures how the linear transforma tion with matrix A changes the volume of a region in R5.
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