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Determine the null space of the following matrix:

$$\begin{bmatrix} 1 & 2 & -3& -1 \\ -2& -4 &6 &3 \end{bmatrix}$$

For this question, I reduced the row echelon form into $$\begin{bmatrix} 1 & 2 & -3& -1 \\ 0& 0 &0 &1 \end{bmatrix},$$ but then I have no idea how to determine the null space, because there's no relationship between $x_1, x_2, x_3, x_4$.

Siong Thye Goh
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Shadow Z
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  • you are encouraged to include your attempt. – Siong Thye Goh Jan 06 '19 at 06:44
  • For this question, I reduced the row echelon form into ( 1 2 -3 -1 0 0 0 1 ), but then I have no idea how to determine the null space, because there's no relationship between x1, x2, x3, x4 – Shadow Z Jan 06 '19 at 06:45
  • Include your attempt in the original post directly. – Siong Thye Goh Jan 06 '19 at 06:46
  • You require $$\begin{bmatrix} 1 & 2 & -3& -1 \
    0& 0 &0 &1 \end{bmatrix}\begin{bmatrix} x_1\x_2\x_3\x_4 \end{bmatrix}=\begin{bmatrix} 0\ 0\0\0 \end{bmatrix}$$ Now just find the relations between $x_1,x_2,x_3,x_4$ as in the answer below.     – Shubham Johri Jan 06 '19 at 06:57
  • See https://math.stackexchange.com/a/1521354/265466 for how to read a basis for the null space directly from the RREF. – amd Jan 06 '19 at 08:13

1 Answers1

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Great that you have found a row echelon form.

From the second row, we can conclude that $x_4=0$.

Also, from there, and the first row of the row echelon form, we have

$$x_1+2x_2-3x_3=0$$

Now you have a relationship between the variables. Hopefully you can take it from here.

Siong Thye Goh
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