Question

1. Consider the following Linear transformation L : R5 + R5 represented in the standard basis via the following matrix: 1 7 4 1 A= 2 4 6 9 -4 0 3 4 3 3 6 12 0 1 9 8 7 9 -2 0 2 (a) Find a basis for Null(A), Col(A), and Row(A). (b) For each v in your basis for Col(A) find a vector u ER5 do that Au = v. (c) Show that the vectors you found in part b together with the basis vectors of Null(A) form a basis for R5. (d) Show that each vector in Row(A) is orthogonal to each vector in Null(A) (When viewing the row vectors as column vectors).

Answer 1

2 3 2. ܢ 3 4 3 4 6 н о 3 о 9 п 12 о -14 2. 4. Ку-ка 13 17: 1: R” - is a linem transformation A. This matrix com be reduced rei echelor form with followsing sequence of trans formations к, -к, +, 8. К--- Ku-3к, - 2 9. 1. ки 3. к - по-че, [o K5 - K, Hчка Ry Rs (scraps) 4. К/, 1). for 5. К - К - 4 к. 12. Ku-K/о 13. K3 - к-ку K, --1, 15. , -к, - 14, К. - 3K, ко 1ч. о Кs + 12 г, 1. Кs 16. К, “к, - « I K - K, - 243 13. K, K, - 25,

After 18th tras formation we obtain O 1 O 1 حم O O . con O 2 After 18th troos formation we we obtain 1 o 1 o Ooooh OOONO 1 0 o o 4 o a tolam is echt aing leading ones and aing leading ones are piret columns . We just use first columns from the original matin 2 ܥ 7 8 wol(A) = 4 6 O 3 — g asis of rowspare. how (A)2 Ningeo lowo in KREF forms (1,0,1,0, 0] [0,1,1,0,0], [0,0,0,1,0], [0,0,0,0,13 S -

o O O A 1 o o 0 0 OOOAA 1 o Lo 1 OO 0 6 by solving this equation we can X=-34, Null(A) 10 this find the null 2, + x₂=0 =) 2,=-X3 2 + 3 = 0 22:-93 = x= x2 X570 so the null spare navedature *420, ngo syane ya = O. 재 2 2 J 2 a -2 where x ER Null (A)= 0 (6) Auzu SS A Alula v 56 39 25 36 36 41 75 Tho 29 35 75 9) 64 85 1165 26 6 18 ] 18 200]

c. is the meetas Civ,+ C₂ V2 a ench Span & An) such that e,,. In Ef fields now take reetus from the list of Null (A) belog span Such that do not to the My and were can create am sine dinenly with auch process which RS. itis independent list spans GRS a basi's

d. if the vectors are orthe i gonal than <Vij vj>?0 for ifj <risvj) to. i=j <,> is the inne product innapuileet <v, w> is lite hom taking the general entidean def of nom. gaverter verter which is given by <ragnicentre ou to low (A). ( (1,0,1,0,0) [0, 1,',00 ] [0,0,0,180 [o, e, 0, 0, 1] T 2VT traspire them 0 (257 5.- 6 o When me mitterity with rulling ***** מ • ** א gives us gew. Implis althogonal