Solving System of Linear Equations: x_2 Free Variable?

[Panphobia]

Homework Statement

|1 0-1-2-8 | -3|
|0 0 1 2 7 | 1|
|0 0 0 1 -4 | -13|


I think that x$_{2}$ is a free variable, so when writing the general solution in a solution set, what would I put for it? Nothing?

 

[Mark44]

Homework Statement

|1 0-1-2-8 | -3|
|0 0 1 2 7 | 1|
|0 0 0 1 -4 | -13|


I think that x$_{2}$ is a free variable, so when writing the general solution in a solution set, what would I put for it? Nothing?

You could say that x₂ is arbitrary or you could say that x₂ = t, an arbitrary real number.

 

[Panphobia]

Oh ok thanks for that, I got it.

 

[Ray Vickson]

Homework Statement

|1 0-1-2-8 | -3|
|0 0 1 2 7 | 1|
|0 0 0 1 -4 | -13|


I think that x$_{2}$ is a free variable, so when writing the general solution in a solution set, what would I put for it? Nothing?

When you write out the equations in detail you see that there is no $x_2$ anywhere in the system. I would not put anything for it; it does not "exist" for this system, no more than $x_{17}$ or $x_{265}$ exist here. However, I suppose you *could* argue the point.

 

[Dick]

When you write out the equations in detail you see that there is no $x_2$ anywhere in the system. I would not put anything for it; it does not "exist" for this system, no more than $x_{17}$ or $x_{265}$ exist here. However, I suppose you *could* argue the point.

I would say $x^2+y^2=1$ in $R^3$ represents a cylinder. Saying it's a circle in the plane because z doesn't occur isn't really a good answer.

 

[Ray Vickson]

I would say $x^2+y^2=1$ in $R^3$ represents a cylinder. Saying it's a circle in the plane because z doesn't occur isn't really a good answer.

I agree, but in the case herein, the "appropriate" view depends on what space you want to operate in.

In $\mathbb{R}^5$ the component $x_2$ is arbitrary, so the described "figure" is like a cylinder parallel to the 2-axis. However, we can look at the problem instead in $\mathbb{R}^3$, where the problem is to represent the (column) vector $(-3,1,-13)^T$ as a linear combination of the 3-dimensional vectors in columns 1--5. The second column is the zero vector, so would have no effect at all on anything in the problem. Having an extraneous $x_2$ does not change the geometry in any way.