Mn(C) → Mn(C) defined by T(A) = A*). (g) The function T : P² →P2 defined by T(f) =f(0)+ f(1)x+ f(2)x². 1.2.4 Determine which of the following statements are true and which are false. *(a) Every finite-dimensional vector space has a basis. (b) The vector space M3.4 is 12-dimensional. *(c) The vector space P3 is 3-dimensional. (d) If 4 vectors span a particular vector space V, then every set of 6 vectors in V is linearly dependent. *(e) The zero vector space V= {0} has dimension 1. (f) If V is a vector space and v1, . span(v1,...,Vn) =V, then {v1,..., Vn} is a basis of .., Vn EV are such that v. *(g) The set {1,x,x²,x',...} is a basis of C (the vector space of continuous functions). (h) For all A E Mn it is true that tr(A) = tr(A"). *(i) The transposition map T: Mm,n →Mn.m is invert- %3D ible. (j) The derivative map D: P5 →P³ is invertible.

Mn(C) → Mn(C) defined by T(A) = A). (g) The function T : P² →P2 defined by T(f) =f(0)+ f(1)x+ f(2)x². 1.2.4 Determine which of the following statements are true and which are false. (a) Every finite-dimensional vector space has a basis. (b) The vector space M3.4 is 12-dimensional. (c) The vector space P3 is 3-dimensional. (d) If 4 vectors span a particular vector space V, then every set of 6 vectors in V is linearly dependent. (e) The zero vector space V= {0} has dimension 1. (f) If V is a vector space and v1, . span(v1,...,Vn) =V, then {v1,..., Vn} is a basis of .., Vn EV are such that v. (g) The set {1,x,x²,x',...} is a basis of C (the vector space of continuous functions). (h) For all A E Mn it is true that tr(A) = tr(A"). (i) The transposition map T: Mm,n →Mn.m is invert- %3D ible. (j) The derivative map D: P5 →P³ is invertible.

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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