1.a) Find a linear transformation L : R³ → R³ whose kernel is spanned by the set()}S =b) Find a linear transformation L : R* → R³ whose kernel is spanned by the set{}S =

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Answered: 1. a) Find a linear transformation L:…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Which of the following mappings are linear transformations? Give a proof (directly using the definition of a linear transformation) or a counterexample in each case. [Recall that PÂ(F) is the vector space of all real polynomials p(x) of degree at most n with values in F.] • (-)-(~`.). = 3y z (ii) : P₂(F) → P4(F) given by o(p(x)) = p(x²) (so o(ax²+bx+c) = ax + bx² + c). (i) 0 : R³ → R² given by 0
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3 be W be vector a positive integer, V and field F, T: VV be linear transformation, and Spaces over f: V²->w be a multilinear alternating function. Let g: V" w be the function defined by Let n a V g (V., ..., Vn ) = + (T(V₁), T(V₂), ...,T (Un)) (V.... . Vn) EV" Prove the following statements. (a g is multilinear and alternating. (b) If V=F", W=F₁, and T=LA for some A € Mnxn (F), then g= (detA/.f
4. Let B = {v1, v2, V3} and C = {w1, w2, w3} be bases for R3, with vectors defined below. -(:) -() --() v1 = V2 = V3 = and --(:) --E) --() 1 wi = , w2 = 1 , W3 = 1 Let L: R³ → R³ be the linear transformation defined by L: Find [L]B and [L]c, the matrices associated to L with respect to B and with respect to C.
Find linear transformation whose kernel is spanned by S.
4. Let VCR" be a subspace. Let F: V→V and G: V → V be invertible linear transformations. Denote by F¹: V→V and G¹: V → V the inverses of F and G respectively. Is the map H: V → V defined by H(v):= F(G(F¹(G¹(v)))), for v € V, a linear transformation? Justify your answer.
5. 3) Let : R₂[x] → R³ be a linear transformation such that ( (1) = = l(x): = 0 and l(x²) = = 1 (a) Find a formula for l(a + bx + cx²). (b) Show that is a vector space isomorphism and find a formula for l-¹.
Let L be a linear transformation from P₂ → R2x2 such that L(f) — ƒ(1) 0 Find dim[Range(L)] ƒ(2) [ƒ(2) ƒ(0)
10. Let L: R R' be a map defined as follows: (4x1 +4 x2 - 2 x3 -9 x4 5x1 +2 x2 - 3 x3-9 x4 6x) – 4 x3 - 9 x4 7x1-2x2-5x3-9 x4) X2 X3 X4, (a) Show that L is a linear transformation on R'. (b) Write out the matrix S which represents L with respect to the standard basis. (c) Find a basis for Range S and Ker S. (d) State the "rank-nullity theorem" and verify explicitly that the result obtained in part (c) matches the statement of the theorem.
4. Let V be a vector space. Prove that a) The zero transformation T(v) = b) The identity transformation T(v) = v for all v EV is a linear transformation. = 0 for all v EV is a linear transformation. CS Scanned with CamScanner

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1. a) Find a linear transformation L: R³ → R³ whose kernel is spanned by the set -{{:) (:)} S =