Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Determine a basis for the solution set of the homogeneous system:

$$\begin{align*} x_1 +x_2 +x_3 &=0\\ 3x_1+3x_2+x_3 &=0\\ 4x_1+4x_2+2x_3&=0 \end{align*}$$

Then the augmented matrix is:

$$ \left[\begin{array}{ccc|c} 1 & 1 & 1 &0\\ 3 & 3 & 1 &0\\ 4 & 4 & 2 &0\\ \end{array}\right] $$

Reduced Row Echelon Form $\to$

$$ \left[\begin{array}{ccc|c} 1 & 1 & 0 &0\\ 0 & 0 & 1 &0\\ 0 & 0 & 0 &0\\ \end{array}\right] $$

I already looked at this example but it didn't help much. I am wondering can someone help to find basis (choosing some parameter for variables $x_1,x_2,x_3$) from RREF.

share|improve this question
add comment

1 Answer

up vote 1 down vote accepted

The first equation in the Reduced Row Echelon Form tells you that we need $x_1+x_2=0$ and the second equation says $x_3=0$. So If we take $x_2=t$ for $t\in\mathbb{R}$ then we must have $x_1=-x_2=-t$ and $x_3=0$.

Thus we have a 1-dimensional solution space determined by the vector

$(x_1,x_2,x_3)=(-1,1,0)$.

This follows from the above discussion since taking $x_2=t$ corresponds to scalar multiplication by t on $(-1,1,0)$.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.