Finding the base and dimension of a system of equations

[Deimantas]

Homework Statement

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

 

[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.

 

[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.

 

[HallsofIvy]

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