Question

Given the matrix A and its reduced row echelon form R, answer the following questions. 4 A 1 0 3 0 3 3 1 1406 1 1 4 1 8 2 1 7 2 13 9 R= 1 0 3 0 3 3 0110 31 000121 1000000 5 Find a basis for the column space of A and the row space of A. Ao Basis for column space of A: with a comma-separated list of vectors enclosed with braces {}. Use vector notation <11, 22,..., Em >. Express your basis using set notation 10 Basis for row space of A a comma-separated list of vectors enclosed with braces {}. Use vector notation <11,12, -, In >. Express your basis using set notation with Calculate each of the following. rank(A) = nullity(A) - nullity (AT) = Find a vector of Ro not belonging to null(A): Use vector notation <11,12, ,16

Answer 1

Here A = 1 O 30 3 3 1 3 3 1 1 406 4 and R= O 1 1 o 1 1 4 185 o 1 2 L o 2 1 7 2 13 9 o 0 оо 0 Where R in now reedured echelon form of A. We th are columna see that in 2nd R and 4 pirot columna. ܝ܇ Basis for column spare of A { [03 pivot rlow ore Again firest. 3 rows of R Hence Basis for now spece of a {(1.05.06 (1,0,3,0,3,3), <51,4,0,6,4), (1,3,4,5,6,573 dim (col spacny) 3 We know, ranka rank A We also have, mullity of A = dim (rusa) (number of non, bivol column] 14 3, mullity for a

We have. rank (A) = rank (AT) manta") = = 3 Now, 1 AT 1 2 1 1 4 4 7 w wo wo h axta on O 2 6 13 45 9 0. AT has 3 pivot column. since ranka = 3 non pivot cdunn AT has (4-3) = 1 > mallity (AT) = 1. 1 Hence multity (=

Now the equation, An=o, is equivalent to Ra = 0 Line mit 3x3 +30s +376 = 0 "2 +83 ting * 016 = 0 4 + 204 +26 = 0 303 325 306 are X2 = a6 az - 3x3 ?!3, ng, do free veriable. 14 = - 205 -76 Now, 2 3 -303-385-346 - 13 - 3x5 -no xz - 245 -16 | 25t ma 1 3 NA 15 as 16 26 Null (A) = Spen འ བ 0 es 10 Let Y = € R & Null (A But V, 6 6