March 3rd, 2017, 04:37 PM  #1 
Member Joined: Nov 2016 From: Kansas Posts: 73 Thanks: 1  Basis
I have 2 vectors v1 and v2 which are linearly independent. I need to find v3 and v4 such that they form the basis of R4. How do I do that?

March 3rd, 2017, 04:39 PM  #2 
Member Joined: Nov 2016 From: Kansas Posts: 73 Thanks: 1 
v1=( 0 1 1 1) and v2=(1 1 2 1)

March 3rd, 2017, 05:35 PM  #3 
Senior Member Joined: Sep 2015 From: USA Posts: 1,944 Thanks: 1011  find a basis for the Null space of your two vectors. Gaussian reducing we get $\begin{pmatrix}0 &1 &1 &1 \\ 1 &1 &2 &1 \end{pmatrix}$ $\begin{pmatrix}1 &1 &2 &1 \\0 &1 &1 &1 \end{pmatrix}$ $\begin{pmatrix}1 &0 &1 &2 \\0 &1 &1 &1 \end{pmatrix}$ $\begin{pmatrix}1 &0 &1 &2 \\0 &1 &0 &3 \end{pmatrix}$ and we can see the Null space is spanned by $\begin{pmatrix}1 &0 &1 &2\end{pmatrix}$ and $\begin{pmatrix}0 &1 &0 &3\end{pmatrix}$ these two vectors along with the original two span $R^4$ as a check you can test that the determinant is nonzero. $\left  \begin{pmatrix}0 &1 &1 &1 \\ 1 &1 &2 &1 \\ 1 &0 &1 &2 \\ 0 &1 &0 &3\end{pmatrix}\right = 12$ 

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