# Thread: General solution in vector form

1. ## General solution in vector form

Hi,

I'm currently trying to understand the general solution to a system of linear equations in vector form. Let's say that when the system is in row reduced matrix form, there is a column will all zeros. According to the image that I took from a textbook, a zero column would suggest the existence of a free variable:

Based on this, can someone please let me know what the general solution in vector form look like? I thought it would be this:

However I'm skeptical since I thought that x3 would be represented somewhere in here, like how x5 was (in other words, I thought that all the free variables were present in the general solution in vector form so I'm confused where x3 is given that x3 is a free variable).

- otownsend

2. ## Re: General solution in vector form

No.

I prefer to use other labels for free variables. Here we would have

$\begin{pmatrix}x_1 \\x_2 \\x_3 \\x_4\\x_5 \end{pmatrix} =\begin{pmatrix}5+3\delta \\ 1+4\delta \\ \lambda \\ 4-9\delta \\ \delta \end{pmatrix}= \begin{pmatrix}5 \\ 1\\ 0 \\ 4\\0 \end{pmatrix} + \delta \begin{pmatrix}3\\4\\0\\-9\\1 \end{pmatrix}+ \lambda \begin{pmatrix}0\\0\\1 \\ 0 \\ 0\end{pmatrix}$

this really highlights the two basis vectors $\begin{pmatrix}3\\4\\0\\-9\\1 \end{pmatrix} \text{ and } \begin{pmatrix}0\\0\\1 \\ 0 \\ 0\end{pmatrix}$

3. ## Re: General solution in vector form

Ok thank you!!!