Linear algebra and Differential equations, Exams of Differential Equations

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JA nal ko ogg pr3 Bld 1O CE MA102 : LINEAR ALGEBRA QUIZ-1 Max Marks: 20 Time: 40 minutes Instructions: 1. Write your Roll No./ Name on your answer script. 2. You will be graded on what you write and NOT on what you intend t your claims and write clearly the appropriate statements for your justifications. 1O write. 3. Justify Question 1: [4+1+2 marks] Consider the homogeneous system of linear equations. _— Ax = 0 with coefficient from R, where ¥ 101 8 1 * 0 -) -5 4-1 6] y_|? _|0 A=], 5 1 5 ipX=ye and 0 =| 9]. : 15270 4 0 ec) ; U ~ form, find all the free variables, Y ~ in the given system. e\ ON (b) Find the rank of A. c NX KS \ (c) Write all the solutions of the given system Ax=0. \ c \ be a vector space. Let A := { Vis V2. ¥3 J}cV be SY 3v2, v2 - 2v1, 3¥1 - V3 }cV (a) By converting A into the reduced row echelon (RRE) Question 2: [5 marks] Let V c R" a linearly independent subset. Is the subset B := { 2v3 - linearly independent? Justify your answer. ] Which of the following subsets V CR? are vector spaces. Find Question 3: [4 marks. fR?. the dimension and a basis of V whenever V is a subspace ol (a) V={ (uy, 2) ERP: 2x + Sy - 11z=0 }. (b) V={ (x,y,0) € R? xyeZ | lab 0) 1c} € Mx3(R) using the Question 4: [4 marks] Find the inverse of the matrix ool Gauss-Jordan method.