Multiple choice

Consider the set of (column) vectors defined by $$X = \left \{x \in R^3 \mid x_1 + x_2 + x_3 = 0, \text{ where } x^T = \left[x_1,x_2,x_3\right]^T\right \}$$.

Which of the following is TRUE?

  1. $\left\\{\left[1,-1,0\right]^T,\left[1,0,-1\right]^T\right\\}$ is a basis for the subspace $X$.
  2. $\left\\{\left[1,-1,0\right]^T,\left[1,0,-1\right]^T\right\\}$ is a linearly independent set, but it does not span $X$ and therefore is not a basis of $X$.

  3. X is not a subspace of R3

  4. None of the above
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B Correct answer
Explanation

X is the subspace of R³ with vectors satisfying x₁ + x₂ + x₃ = 0, which is a 2-dimensional subspace. The given vectors [1,-1,0]ᵀ and [1,0,-1]ᵀ both satisfy the equation and are linearly independent (not scalar multiples). However, they don't span X - for example [0,1,-1]ᵀ is in X but cannot be expressed as a linear combination of just these two vectors. A basis for X would need to be more carefully chosen or have different vectors.

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