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Question 2 | 13 points Show details of your work. [1 -2 0 0 3 [1 0 0 -2 37 2 -5 -3 -2 6 0 1 0 -1 0 Given matrices 0 5 15 100 and 00110 (2 6 18 8 6 0 0 0 0 0 where the matrix R is the reduced echelon form of the matrix A. a) [2 points Find a basis of the Row(A) {{1,0,0,–2,3], [0,1,0, -1,0], [0,0,1,1,0}} b) [1 point) Find the Rank(A) = 3 c) [2 points Find a basis of the Col(A) {{1,2,0,2)", [-2,-5,5,6)", [0, -3,15,18)"} d) [1 point Find the Nullity(A) - 2 e) [2 points Find a basis of the Null(A): Solve AX-0, i.e. solve RX-0 x = 2x, - 3x ; x = x ; x =-X, ; X, and xs are free variables Hence, a basis of Null(A) is {(2,1,-1,1,0) (-3,0,0,0,1)"} [ 3 points Write each linearly dependent column of A as a linear combination of the linearly independent columns of A Linearly DEPENDENT columns of A are the column 4 and the column 5. Linearly INDEPENDENT columns of A are the column 1 column 2 and the column 3. Column 4 --2 Column 1 of A - Column 2 of A + Column 3 of A Column 5 = 3 Column 1 of A g) [2 points Find a basis of the Col(R) {[1,0,0,0], [0,1,0,0], [0,0,1,0) Page 3 of 6

Answer 1

We have A 0 0-2 31 1-2 00 37 12 -5 -3 -2 6 and Rolo 0, 0 - 0 10 5 15 10 ol Too , i o 2 6 18 8 6 / To o o o o echelon form of the matrix A. the we apply row operations of A get 1 -2 0 0 3 2 -5 -3 -2 6 o 5 is 10 o 2 6 18 8 61 -2003 -2 -3 -2 0 5 15 100 -). R2 1-2 0 0 37 R T-2 0037 6 | 3 3 2 1 To 32 0 10 1 3 2 0 1 0 0 0 0 0 10 10 18 8 of 0 -12 -12 0 - I.Rai -2 0037 R₂-3Rg I -2 0 0 37 2 013 20- 0 1 0 -1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 R. RO I -2 0 0 37 To , o o 10 o 1 I ol I 0 0 0 0 0 КамА row non-zero 4ure the row space 30 a basis of RONA. WA {(1, 2, 0, 0,3), (0, 1, 0,-1,0), onalano-10 10,0,1,1,0) of son space (6) Rank (4) - Dimension

sing column operations on A, we have -2 - 5 o - 15 o - 3 7 ( о , о 24 ) o o o "- з -) - 3 - 2 5 15 10 | 2 10 х 6 o 7 O о о | Г) o 2 - отия o -) 2 Г) o o o o 7 | 2 оо - о | o o o 5 о 2 6 5 1 of 10 1 о Г 2 o | 2 . c, -2 С, o o o o 7 o o -) о o o 5 о 1 5 4 0 1 . o o o o 7 | 2 o o - о | o o o 5 о І о то оо c, C4 | 1 o o o on | 2 0 - o o/ оо 5 o o | 2 Р. Lol oool A bases of a col(A) & the non-tuo columns of P. 27,2, 0, 0), (0, 0, 0, ) 7, 1,5,0)/ we know Rank(A) & Nullty (4) - dimension s, а он дел 4. substra 1) - 3 + 4) - 5 е роth писh (9) г. in les your ans in p, " - sr, sn't 2 Г) - 3 0 -5 - 3 1 0 5 15 12 6 18 1 Г) o 3 3 | 2 - 29 | o 5 1 1 10 6 - 1 5 ) | І тоо , 4 o - H) 3 o- По о 5 | 0 0 1 6 / 1 2 1 L 4 Ром! - ооо Ал о

26 - و 7 6 0 اا ا- و 3) / 0 رہ ا و /