Question

1 (8 pts) Find the dimension and a basis for the following vector spaces. (a) (4 pts) The vector space of all symmetric 2 x 2 matrices (which is a subspace of M22). (b) (4 pts) All vectors of the form (a, b, 2a +36) (which is a subspace of Re"). 2. (12 pts) Given the matrix in a R R-E form: -21 1 [1 0 0 0 3 0 1 1 0 - 2 0 0 0 1 0 0 0 0 0 1 1 0 (a) 6 pts) Find rank(A) and millity(A), and nullity(A). (b) (2 pts) Find a basis for the row space of A. (e) (2 pts) Find a basis for the column space of A. (d) (2 pts) Find a basis for the null space of A.

Answer 1

1: (a) The vector space of all symmetric marices. 2*2 v={ AE Mz2 = A=A a=a7} AL d] than at Let A= d Since A= AT :, boc Then -=[:] [:] =a[:] +62 1 td

basis for this vector space is {[8], 1.1 : ] Hence the dimension of the vector space is 3. (6) V= (a, b, 2a+3b)R? a, bEIR (a, b, 2a +36) = a (1,0,2) +b(0,1,3) a basis for this vector space is {(1; 0, 2) 2), (0,1,3 3 ) ] Hence the dimension of the rector space is 2. 2. The matrix in R.R-E form o 3 -2 - 2 A= 0 0 0 0 0 o 0 ca] pank (A) = number of non-zero rows in A =3

we know that, rank (A) + nullity (A) = number of column of A TA 3 + nullity (A) = 6 (A) = 6-3 = 3 nullity Now ATE o O 0 o 0 3 -2 o -2 0 we know that rank (A) = pank (AT). 3 = pank (AT) :. rank (AT) = 3 Again, rank (A") + nullity (AT) nullity (AT) = number of column of AT + nullity (AT) = 4 nullit (AT) = 4-3 = 1, » 3 + > lity （2） a basis for the row space of non-zero pows of A . is consisting three lo SO basis 9 0 -2 3 -2

o (C) 3 2 Oo .2 As o o 0 O o O 0 C3=C3-2 c'4 = cq + 2C2 26-06-02 0 3 -2 00 o O o o 0 O O © cs = C5-36 C%= 6+261 O O O o ch=Cg - cu co=C6-C4 0 O column reduced eachelon form o 0 o O 0 Basis for the column space of A consis three non- non-zero column of A. Basis 7711453

(4) for basis of the null space of A. З 2. 21 о -2 х, О о Х3 хч О О О О Ge 9x Then And х1 +375 - 2 X6 = O + х = -3х5 +2 х. х2 + 22 - 2 Хs+ % o=x2 = – X3+2х5 - 36 х4+xs + +“, 74 = -35 -х And - 74 le Х). Х. X4 X5 -350X6 - Х3 +2X5 - Хs ха 9x - 5х- 26 X5 26 З хз 2. +25 det от- о стото: о - о a basis for null space IS 2. 9. 1 о

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