Question

For Problems C4-C11, prove or disprove the statement. C4 If V is an n-dimensional vector space and {11,...,Vk} is a linearly independent set in V, then k sn. C5 Every basis for P2(R) has exactly two vectors in it. C6 If {V1, V2} is a basis for a 2-dimensional vector space V, then {ağı + bū2, cũı + dv2} is also a basis for V for any non-zero real numbers a,b,c,d.

Answer 1

The given statement is true y If Vis an n-dimensional vector space and ca, ta, ...? set in V. then ksn. [True] is linearly independent Since dim v=n. then let Iroop {appe, th} be a basis of v. Since proces, NE V. and vi 70, Then V = 0, a tl₂q2 t. tenan where c,,Cr, ...en Zero let cito are scalars. not all of which are By Replacement theorem .Yi-1, V2, Pitt and is a basis of v. 24" and Nq=d, ait daar tot di-,&; -, Let tdip + diti &; to I dnan where di are seal ar not all zero

hon zero We assert that at least on of d,, dz, -- di 1, di, dito, .. do is Because if all of them be Zero then V₂ = divi this will imply linear dependence of V evhich va contradiction let dj to it. By Replacement M theorem {d,,dz,--di-1, Y, diti, ...jajai, aj ty dn) is a new basis of v. Since Vz fo.

V N21 "3=t, &, + t₂l2t ...tti-,&;-it telttitoditi t; -,2;_ , + tj V₂ + titi Kita +. t - ttaan. where ti scalar not all Zero a non Zow. We assert that at least one of to, ta, ... tn is Because otherwise V3 = ti av, t t; v2 and this will imply a contradiction. Proceeding in this away we are observe that each step is replaced by one Vi and the one di

set remains a basis of v. cases may arise. resulting The The following VV2, basis containing o all come to the new some li s. In this Case к{n . a; s and V, V2, Vx exhaust all the new basis In this case form K-n. all after another becomes a and qvium in] It can not happen that kyn. Because then on by Replacement therrum n rectors (1, ...) will come to the basis replacing new basis of v. Therefore the remaining rectory each "ht, Unt2, - k of u linear Combination of V.Vn that the set {", 42, VA} As linearly dependent a contradiction Therefore isn. This complete the prop. ca since dimp(P) = 3. Then will be each a Show its basis contain exactly 3 vectors cute linearly

which is c? Since dim P (IR) = 3. Then its basis contain exactly 3 veeteris linearly independ cat-vectoren. for example torre {1,2,2%) and contain three independent veeters, Hence the given statement is false. a basis of P₂ (12) linearly als is 2. co The statement is false take R2 whose dimension Now { (4,0),(0,1)] basis of IR". then (10,9 +(0,1), 2(,0) +2(0,017 {(1,1), (2,2,3 linearly dependent which arre Hence Note: ! (2, 2) = 2.(1/1). {(1, 1), (2,2)) is not a basis of 1R² Replacement theoytem ! IF (D, Ag, -an] be a basis of a vector field and non-zou veetore ß of v p=cate dat toan if estoqa,, . dj.., P, "ite, , tn) is a new o over Space v is expressed Ciff then au basis of v