Consider the set (column) vectors defined by X = { x in R^3 | x1 + x2 + x3 = 0, where x = [x1, x2, x3]^T }. Which of the following is TRUE? A. { [-1,0,1]^T, [1,-1,0]^T } is a basis for the subspace X. B. { [-1,0,1]^T, [1,-1,0]^T } is a linearly independent set, but it does not span X and therefore is not a basis of X. C. X is not a subspace of R^3. D. None of the above.

GATE 2007 · Engineering Mathematics · Vector Space · medium

Answer: A. { [-1,0,1]^T, [1,-1,0]^T } is a basis for the subspace X.

1. Verify dimension of X: X is defined by one equation x1+x2+x3=0, so the coefficient matrix A = [1,1,1] has rank 1. Thus dim(X) = 3-1 = 2.
2. Check both vectors belong to X: For [-1,0,1]^T: (-1)+0+1=0. For [1,-1,0]^T: 1+(-1)+0=0. Both lie in X.
3. Check linear independence: Suppose c1*[-1,0,1]^T + c2*[1,-1,0]^T = 0. Row 2 gives -c2=0, so c2=0. Row 1 then gives -c1=0, so c1=0. The vectors are linearly independent.
4. Conclusion: Two linearly independent vectors lying in the 2-dimensional subspace X span X. Therefore S = { [-1,0,1]^T, [1,-1,0]^T } is a basis for X.