Question

Let {(1, 1, 1, 1), (1, 2, 1, 2)} be linearly independent which is a subset of vector space V₄. Extend it to a basis of V₄

Answer 1

In vector space $V_4$ any four linearly independent vectors form a basis. Let us extend

$\{(1, 1, 1, 1), (1, 2, 1, 2)\}$ to $\{(1, 1, 1, 1), (1, 2, 1, 2), (0,0,1,0),(0,0,0,1)\}$.

Since $\left|\begin{array}{cccc} 1 & 1 & 1 & 1\\ 1 & 2 & 1 & 2\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end{array} \right|= \left|\begin{array}{cccc} 1 & 1 & 1 & 1\\ 0 & 1 & 0 & 1\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end{array} \right|=1\ne 0,$ we conclude that the vectors

$(1, 1, 1, 1), (1, 2, 1, 2), (0,0,1,0),(0,0,0,1)$ are linearly independent, and thus form a basis of $V_4.$