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Can anyone explain how to do this? Usually when the question gives a vector and asks whether it is in span with some other vectors I put them in a matrix and calculate the determinant and so on but with this question type I am not really sure

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2 Answers

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To span $\mathbb{R^3}$ you need 3 linearly independent vectors. You can determine if the 3 vectors provided are linearly independent by calculating the determinant, as stated in your question.

If you have 3 linearly independent vectors that are each elements of $\mathbb{R^3}$, the vectors span $\mathbb{R^3}$.

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Option (i) is out, since we can't span $\Bbb R^3$ with less than $\dim \Bbb R^3 = 3$ vectors. If you have exactly $\dim \Bbb R^3 = 3$ vectors, they will span $\Bbb R^3$ if and only if they are linearly independent -- for this reason it suffices to check determinants. Compute $$(ii) = \begin{vmatrix} 1 & 1 & 1 \\ 1 & 0 & 2 \\-1 & -4 & 2 \end{vmatrix}\quad\mbox{and}\quad (iii) = \begin{vmatrix} 1 & 0 & 1 \\ 0 & 1 & 2 \\ -1 & -4 & -2\end{vmatrix}.$$One should select options (ii) and/or (iii) in the statement of the problem if and only if the corresponding determinants above are nonzero.

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