# Finding the basis given a system of equations in R4

The question I'm given is this:

Let $S$ be the subspace of $R^{4}$ consisting of the solutions to the following system of equations:

$$x_{1}+2x_{2}+2x_{3}+2x_{4}=0$$

$$x_{1}+5x_{2}+8x_{3}+5x_{4}=0$$

$$-3x_{1}-4x_{2}-2x_{3}-3x_{4}=0$$

I am asked to give a basis for $S$.

I have attempted to solve this by putting this into a matrix and solving for my $x_{1}$, $x_{2}$, $x_{3}$, and $x_{4}$, but I don't really know where to go from there. I am sure I'm missing something important.

Any help would be greatly appreciated!

• What is the rank of associated matrix?
– user261263
May 25 '16 at 8:07
• I believe the rank of the matrix is 3. May 25 '16 at 8:13
• And the principal minor?
– user261263
May 25 '16 at 8:14
• I don't quite understand how to find the principal minors. Is it just taking the determinant? May 25 '16 at 8:22

## 2 Answers

You can also do it without matrices, simply by solving the equations. Label them (1),(2),(3).

4(1)-(2): $3x_1+3x_2+3x_4=0$ (4). Then -(2)-4(3): $11x_1+11x_2+7x_4=0$ (5). Now 11(4)-3(5): $x_4=0$ and so $x_1+x_2=0$. (2) now gives $x_2+2x_3=0$. So putting $x_3=k$ we have $x_1=2k,x_2=-2k,x_3=k,x_4=0$.

Check: (1) $2k-4k+2k=0$, (2) $2k-10k+8k=0$, (3) $-6k+8k-2k=0$.

So a basis for $S$ is $(2,-2,1,0)$.

You are on the right way. Rewriting this question using matrices becomes: Find a basis of the kernel of $$\begin{pmatrix}1&2&2&2\\1&5&8&5\\-3&-4&-2&-3\end{pmatrix}.$$ You can either solve this like @almagest suggested or you can use some more soffisticated techniques. In this example I suggest to perform a row reduction. How to do this is explained here wikibooks.

• If you struggle you can add a comment and I will help you through the steps. May 25 '16 at 8:33
• So I have solved the matrix to find x1=2t, x2=-2t, x3=t, x4=0. What do I do with this after I find the solutions? May 25 '16 at 8:45
• Solved the matrix is not the right term but you are done. You have written the solution as 'x=tv' with 'v=(2,-2,1,0)' hence the solution space is one dimensional with basis $\{v\}$. May 25 '16 at 9:10
• Thank you, the answer was there but I was thinking too complex for this question. May 25 '16 at 14:06