Question

Let { (1,1,1,1) , (1,2,1,2) } be a linearly independent sbuset of a vector space V4. Extend it to a basic of V4

Answer 1

It is enough to find a matrix $L=\left(\begin{array}{llll}1&1&1&1\\1&2&1&2\\x_1&x_2&x_3&x_4\\y_1&y_2&y_3&y_4\end{array}\right)$ wit nonzero determinant. Set $x_1=0$, $x_2=0$, $x_3=1$, $x_4=0$; $y_1=0$, $y_2=0$, $y_3=0$ , $y_4=1$. The determinant of the matrix is: $det(L)=\det(A)\det(B)$, where $A=\left(\begin{array}{ll}1&1\\1&2\end{array}\right)$ and $B=\left(\begin{array}{ll}1&0\\0&1\end{array}\right)$. $det(A)=1$, $det(B)=1$. $det(L)=1$. Thus, the basis is: $(1\,\,1\,\,1\,\,1)$, $(1\,\,2\,\,1\,\,2)$, $(0\,\,0\,\,1\,\,0)$, $(0\,\,0\,\,0\,\,1)$.