Linear Algebra

198 Chapter Three. Maps Between Spaces(b) Where does the transformation send this vector?(05)(c) Represent this transformation with respect to these bases.( ) ( ( ( )1 1 2 −1B = 〈 , 〉 D = 〈 , 〉−1 1)2)1(d) Using B from the prior item, represent the transformation with respect toB, B.1.23 Suppose that h: V → W is nonsingular so that by Theorem 2.21, for anybasis B = 〈 ⃗ β 1 , . . . , ⃗ β n 〉 ⊂ V the image h(B) = 〈h( ⃗ β 1 ), . . . , h( ⃗ β n )〉 is a basis forW .(a) Represent the map h with respect to B, h(B).(b) For a member ⃗v of the domain, where the representation of ⃗v has componentsc 1, . . . , c n, represent the image vector h(⃗v) with respect to the image basis h(B).1.24 Give a formula for the product of a matrix and ⃗e i , the column vector that isall zeroes except for a single one in the i-th position.̌ 1.25 For each vector space of functions of one real variable, represent the derivativetransformation with respect to B, B.(a) {a cos x + b sin x ∣ ∣ a, b ∈ R}, B = 〈cos x, sin x〉(b) {ae x + be 2x ∣ ∣ a, b ∈ R}, B = 〈e x , e 2x 〉(c) {a + bx + ce x + dxe x ∣ ∣ a, b, c, d ∈ R}, B = 〈1, x, e x , xe x 〉1.26 Find the range of the linear transformation of R 2 represented with respect tothe standard ( ) bases by each ( matrix. )( )1 00 0a b(a)(b)(c) a matrix of the form0 03 22a 2b̌ 1.27 Can one matrix represent two different linear maps? That is, can Rep B,D (h) =Rep ˆB, ˆD(ĥ)?1.28 Prove Theorem 1.4.̌ 1.29 Example 1.8 shows how to represent rotation of all vectors in the plane throughan angle θ about the origin, with respect to the standard bases.(a) Rotation of all vectors in three-space through an angle θ about the x-axis is atransformation of R 3 . Represent it with respect to the standard bases. Arrangethe rotation so that to someone whose feet are at the origin and whose head isat (1, 0, 0), the movement appears clockwise.(b) Repeat the prior item, only rotate about the y-axis instead. (Put the person’shead at ⃗e 2 .)(c) Repeat, about the z-axis.(d) Extend the prior item to R 4 . (Hint: ‘rotate about the z-axis’ can be restatedas ‘rotate parallel to the xy-plane’.)1.30 (Schur’s Triangularization Lemma)(a) Let U be a subspace of V and fix bases B U ⊆ B V . What is the relationshipbetween the representation of a vector from U with respect to B U and therepresentation of that vector (viewed as a member of V ) with respect to B V ?(b) What about maps?(c) Fix a basis B = 〈 β ⃗ 1 , . . . , β ⃗ n 〉 for V and observe that the spans[{⃗0}] = {⃗0} ⊂ [{ ⃗ β 1 }] ⊂ [{ ⃗ β 1 , ⃗ β 2 }] ⊂ · · · ⊂ [B] = V