# GATE | GATE-CS-2007 | Question 27

• Difficulty Level : Hard
• Last Updated : 22 Aug, 2019 (A) A
(B) B
(C) C
(D) D

Explanation: To be basis of subspace x, 2 conditions are to be fulfilled

1) They must span x
2) The vectors have to be linearly independent

Attention reader! Don’t stop learning now.  Practice GATE exam well before the actual exam with the subject-wise and overall quizzes available in GATE Test Series Course.

Learn all GATE CS concepts with Free Live Classes on our youtube channel.

1)the general solution of x1+x2+x3=0 is [-x2-x3 , x2 , x3]^T (Transpose)

Which gives two linearly independent solutions by assuming x2 = 1 and x3 = 0 and next x3 = 1 and x2 = 0 gives [-1,1,0]^T and [-1,0,1]^T respectively. Since both of these can be generated by linear combinations of [1,-1,0]^T & [-1,0,1]^T given in question, it span x.

2) Above set of column vector is linearly independent because one cannot be obtained from another by scalar multiplication
(second method rank is 2..that is why linearly independent)

Quiz of this Question

My Personal Notes arrow_drop_up