Question

1. Find A if (2A)' = [ : :] 2. Determine if {(x,y) : x - y = 1) is a subspace of vector space V - R 3. Let vi, V2, V3 be three linearly independent vectors in a vector space V. Is the set {v1 - 2v2, 2v2 - 3v3, 3V3 - Vi} linearly independent or linearly dependent? Prove your answer.

Answer 1

1 (2A)' 2 O 4 1 Inverse both sides, on Taking (12A)"] 1) 28 - zo lo 1 -4 2 -b 1 a с b d d ad-bc I-C a 2A- 1 1 2 -4 2 O 11 -4 2 4 A. 1/4 -1 1/2 2) is {(x,y) Vector space x-y = 1 a subspace of IR?

is o subset w of Vector Space v V if and only if a subspace of 1.) OE W W is closed 2) vector under " addition then witw, if w, and also belong to wz belong to w W 3 W is W is closed under scalar Multiplication if and real number then cw E W. WE W с. is a Here W = {1x, y): x-y = 1} V = IR ? W is Not Subspace of IR² because 1) Zero Vector (0,0) doesnot belong to W. 2.) W Not with respect to Vector Addition JS Closed (2,1) EW Now Not (3,2) E W (291) +13, 2) (5,3) ¢ W : 5-3-2) 3. W is Closed under Scalas Multiplication (2,1) EW let C= 2 212,1) (492) & W ( 4-2 = 2) So {1x,y) : x - y = 1} is Not a subspace of IR?

3) Given :- ا و ا وا independent Vectors Now are linearly in a Vector space V. To find out = $0-2uz , 2 vş -343, 3 v3 - 6;} is linearly dependent or Independent. fu,,V21 A set of Vectors on} is Linearly Independent of the vector equation civitat tcnun=0 has only the trivial solution G=0, C2-0, If there exist scolars CigC 29 not all equal to o such that ciuit CzUzt...t covn =D then {o,guza Linearly Dependent. cn=0. Cn معا و is 9 So let there exists scalars Cg C2, C3 such that 110-2v) + (2 (202-303) + (3 (303 - ) - 0 Gų - 20,V2 + 24V, - 3c,v + 3cz6z - Tzu - 0 vi 19 - C3) + (-26, +2cm) + 13 (-362+363) = 0 But şaig 929 93} are linearly independent 9-C3=D ? G = C3 G = = -24, +2c2-0 C3 C=C2 - 3C2 + 3G=0 C2 = C3 Since 9,629 (z can be any real no. (they can be non zero also). So {u, -2 0z , 262,363, 3v3 -8;} is Lineadly Dependent