Question

(10 points) Consider the following system of linear equations. 2x1 + 4x2 - X3 = 0 31 +2302 + x3 = 3 (a) Write the system as a vector equation in which the left-hand-side is a linear combination of column vectors. (b) Find the solution set of the system in vector form. Check that every solution is the sum of a particular solution and a vector in the null space of the coefficient matrix. (c) Find a basis for the null space of the coefficient matrix.

Answer 1

اول a ar 7 0 Given 2717472-7350 kr + 2x2 + 2y=0. Giren System can be Written ait le jazt (1) az (real us apply elementary operations on matrin. coeff. (? 기 --) (7)« 6 res the 2 R12 니 12;( ( 2 4 2 - 2 22-224 2 --/3R 0 0 - 3 R-P2 2 0 o 0 The system in equivalent to Riten D nz FO to het R2F.C. Ľ CER, arbitrary) Then y=-20, -2c, nz -o DUHENNT DERANCE

M u 3-6133= [3] nz Rea. Solution set in r. 3 [?]:ceiry creay we nee noin that every in the hull space of the coefficient matrix. Because En null space is 2 X AX=0 į CA in the strin) 3<[%]:ceres coefficient maitrin Here it in с An null pace in ?X: AX = o hence Null (A) span and the net ? [7] } in hence basino for the null space of coeff. matrix is_ a linearing independent set 3 [3]