# Homework Help: Finding the base and dimension of a system of equations

1. Feb 26, 2012

### Deimantas

1. The problem statement, all variables and given/known data

Find the base and dimension of a system of equations:

3x1 - 5x2 + 2x3 + 4x4 = 0
7x1 - 4x2 + 1x3 + 3x4 = 0
5x1 + 7x2 - 4x3 - 6x4 = 0

2. The attempt at a solution

Written in matrix form:

3 -5 2 4
7 -4 1 3
5 7 -4 -6

What I get:

11 -3 0 2
7 -4 1 3
11 -3 0 2

so, that's two identical rows. That means there's an infinite number of solutions, I think. What gives? I'm lost. Any help to solving this exercise is appreciated

2. Feb 26, 2012

### HallsofIvy

What do you mean you "got" that? How did you get it? Did you do a row reduction? Typically one does a row reduction to get the first column
$$\begin{bmatrix}1 \\ 0 \\ 0 \end{bmatrix}$$

Also, you don't mean "find the base and dimension" of the system of equations. You mean to find the dimension of the solution set of the system of equations.

3. Feb 26, 2012

### Deimantas

I'm not very good at this subject, as you can see. Yes, I did row operations on the matrix. I just wanted to know if having two identical rows in a matrix can help in solving this kind of exercise, but I guess not.

Maybe this would look clearer?:

3 -5 2 4 = 3 -5 2 4 = 3 -5 2 4 = 1 -5/3 2/3 4/3
7 -4 1 3 = 21 -12 3 9 = 0 23 -11 -19 = 0 23 -11 -19 = 0 23 -11 -19
5 7 -4 -6 = 15 21 -12 -18 = 0 46 -22 -38 = 0 23 -11 -19 0 23 -11 -19

Is this correct? What next? Like I said, I'm not good at linear algebra

I can't believe it, the whitespace seems to be ignored in the posts... The answer is in bold, a two rows, 4 columns matrix.

Last edited by a moderator: Feb 26, 2012
4. Feb 26, 2012

### HallsofIvy

Okay, so that reduces to
$$\begin{bmatrix}1 & -5 & 2 & 4 \\ 0 & 23 & -11 & -19 \\ 0 & 0 & 0 & 0\end{bmatrix}$$