Linear Algebra

318 Chapter Four. Determinantš 1.21 What is the volume of a parallelepiped in R 3 bounded by a linearly dependentset?̌ 1.22 Find the area of the triangle in R 3 with endpoints (1, 2, 1), (3, −1, 4), and(2, 2, 2). (Area, not volume. The triangle defines a plane — what is the area of thetriangle in that plane?)̌ 1.23 An alternate proof of Theorem 1.5 uses the definition of determinant functions.(a) Note that the vectors forming S make a linearly dependent set if and only if|S| = 0, and check that the result holds in this case.(b) For the |S| ̸= 0 case, to show that |T S|/|S| = |T | for all transformations,consider the function d: M n×n → R given by T ↦→ |T S|/|S|. Show that d hasthe first property of a determinant.(c) Show that d has the remaining three properties of a determinant function.(d) Conclude that |T S| = |T | · |S|.1.24 Give a non-identity matrix with the property that A trans = A −1 . Show thatif A trans = A −1 then |A| = ±1. Does the converse hold?1.25 The algebraic property of determinants that factoring a scalar out of a singlerow will multiply the determinant by that scalar shows that where H is 3×3, thedeterminant of cH is c 3 times the determinant of H. Explain this geometrically,that is, using Theorem 1.5,̌ 1.26 Matrices H and G are said to be similar if there is a nonsingular matrix Psuch that H = P −1 GP (we will study this relation in Chapter Five). Show thatsimilar matrices have the same determinant.1.27 We usually represent vectors in R 2 with respect to the standard basis sovectors in the first quadrant have both coordinates positive.( )⃗v+3Rep E2(⃗v) =+2Moving counterclockwise around the origin, we cycle thru four regions:( ) ( ) ( ) ( )+ − − +· · · −→ −→ −→ −→ −→ · · · .+ + − −Using this basisB = 〈(01),( )−1〉 β1 ⃗0β2 ⃗gives the same counterclockwise cycle. We say these two bases have the sameorientation.(a) Why do they give the same cycle?(b) What other configurations of unit vectors on the axes give the same cycle?(c) Find the determinants of the matrices formed from those (ordered) bases.(d) What other counterclockwise cycles are possible, and what are the associateddeterminants?(e) What happens in R 1 ?(f) What happens in R 3 ?A fascinating general-audience discussion of orientations is in [Gardner].1.28 This question uses material from the optional Determinant Functions Existsubsection. Prove Theorem 1.5 by using the permutation expansion formula forthe determinant.