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March 3rd, 2017, 05:37 PM   #1
ZMD
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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?
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March 3rd, 2017, 05:39 PM   #2
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v1=( 0 1 -1 1) and v2=(1 -1 2 1)
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March 3rd, 2017, 06:35 PM   #3
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Quote:
Originally Posted by ZMD View Post
v1=( 0 1 -1 1) and v2=(1 -1 2 1)
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 non-zero.

$\left | \begin{pmatrix}0 &1 &-1 &1 \\ 1 &-1 &2 &1 \\ 1 &0 &-1 &-2 \\ 0 &1 &0 &-3\end{pmatrix}\right| = 12$
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