Question

[3 1 0 -1] [3 1 0 -1] Same A = 3 1 -7 1 , and its row echelon form is U = 0 0 -7 2 . 16 2 0 -2] LO 0 0 0 ] 1. If C(A) is to be a subspace of some Rk, what is k? Is C(A) a subspace of your chosen Rk? Why or why not? 2. A revisit to Problem #5 of Homework 06. (a) Observe that in matrix U, column-2 is a scalar multiple of column-1. Does the same relation occurs in matrix A, always, sometimes, never? Explain. (b) Also observe that in matrix U, column-4 is a linear combination of column-1 and column-3. Does the same relation occurs in matrix A, always, sometimes, never? Explain. (c) True or false: C(A) = C(U)? Explain why.

Answer 1

olution an 14 A=131 0-17 te echleontoom is 3 lit 4. te U3 10-17 O 0- 72 LO 0 0 0 since columns of A are vectors in R. If ((A) is to subspace of Irk for somek, it will be of R² . As un echleon from U of A, bivot is present in Ist 4 3rd column coresponding columns of matrix A forms basin of C(A). "C(A) has basis & (3). 189. (8) = 0(1) +0(1) :: BECCA) ".((A) is non empty let u,ve CIA) Then u= al 31. tu= ( 1 ) + d 12 for some scolars utve a (3) ***)+()+ a(3) = (ate) ( ) +(6+d) () :. utv EC(A) Nau, for any scalan t, conaidu tu-t (913+1%8)) til 3 ft (3) ituecia)

As all conditions of subspace are satisfied CLA) is subspace of Rs. Q3 a) In matrix o, column 2 a) In matut is scalar multiple of columns 18 = + 1) occur in matrix A always same relation will co as elementary now operations does not change the relation blw columns of matrix. I we can check also (!) = 73 (3) b) In matix u, column 4 is linear combinator of column I and column 3 as 13)=313)315) IÓ I lol !! The same relation will occur in matrix A always as elementary new operations does not change relation b/w columns of matrix We can check ako 7) -+) (3) mo I -2

amal “) Tone ((A) = Clu) As elementau elementary now operation does not cho column space of matux. sis for c(0) = { 13.2013 as pivot is 3) Basis for c(u) = (%) (-3) J present in 1st 4 lllrd column. Coresponding columns of matrix A will from basis for CLA) : Basin fou com a ( 1), (3)).