Question

linear algebra question easy, please answer fast with steps

Mark each statement True or False. Justify each answer. Here A is an mxn matrix. Complete parts (a) through (e) below a. If B is a basis for a subspace H, then each vector in H can be wrben in only one way as a linear combination of the vectors in B. Choose the correct answer below O A. The statement is false. Bases for a subspace H may be linear dependent and therefore there can be mutple solutions for the same vector xin H O B. The statement is true. All bases for a subspace H are ineary independent and therefore each vector in H can only be generated as one unique inear combination of the vectors in 8 O C. The statement is true. Suppose B Vv and i is a vector in H that can be generated teo ways. Say, xc s, and xd, +4, then 0x-x(-d)-(-4)% Therefore, c=d, and x can only be generated in one way O D. The statement is false. Suppose B ) and x is a wector in H. The vector x can be generated in a muliple of ways based on the values of the vectors in the set B ) v, b. If 8- (V-Y is a basis for a subspace H of R, then the correspondence x-xle makesH look and act the same as RP. Choose the corect answer below The statement is false. The vectors in H may contain more than p entries and therefore ther correspondence x-xla does not make H look and act the same as RP O A. OB The statement is true. The correspondence of x-xe implies a one-to-ane corespondence between H and RP that preserves linear combinations makes O D. The stabement is true. The fact that 8v,.3 imples that the comespondeence x-l O C. The statement is false. The correspondence of x-xa does not imply a one-to-one H look and act the same as RP correspondence between H and RP that preserves linear combinations c. The dimension of Nul A is the number of variables in the equation Ax 0. Choose the correct answer below O A. O B. The statement is false. The dimension of Nul A is the number of variables in the equation Ax 0 minus the number of free variables in the equation Ax 0 The statement is false. The dimension of Nul A is the number of free variables in the equation Ax 0

Answer 1

@ If B is a basis for a subspace H. Then each vector in H can be written only One way as a lincar combination of the vectors in B. Any @ The statement is true. Because, Bose, U2,.., dp) bo is a basis for I and det. Suppose, a can be generated in two way say ascio, te det... tcp op again ne dil, of d2025-. tdp Up : 0= (^_^) Ce-d JV, 4 (22-d.) Vet.. +(Cp-dp) Up Since, soi,...,Up] are linearely independent . Grdico s cedi C2-0220 C2ad2 Cp-dpzo Sprdp. -. x can be generated only in one way. b) If Bobs 01.02.., Op} is a sobspace Ĥ of Rn, then the corresandmee. as [2]r makes it look and act. the same as IRP

Ars A The statement is true. The correspondente Tan [n]g implies a one-to-one correspondence between it and RP that preserve linar combination. Because, nac, 0, +2024. tops det, [x]B = [M]B d) Again? HIM! sci-d, czadz, ... Cpadp. for any per ERP ya eid teiden 4 pp. preserves tiras combinant . It is bijective of [cardy]g ca] p + a[y]B. is o of Null A the number The dimention of An=0. variables in the equation A The Statement is false. The dimention vot Null A is number of free variables in the equation And 0. example eur, A ziani 124-1)

Augmented marrin ri 1 31.07 2 ss o We are LI24-1) Shag Mo 21 310 7 w - - 0 ah - For An 20 are here, 21+223 +3^420 1164120 92 +23-2ny 20 :. 212 - 223:32 faza-23724 1-293-324 11- 123 12. : Null Space A La -2 :. dimention of Mull Az 2 = No of free variable o The dimension of the Column Space of A is rank A suncio AMC The Statement is true. The is rank of A the dimention of Column Space A. O