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Dekoyl's Questions

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dekoyl:
Just a quick one:
I'm given three linear equations. Using matrices, I find that it's in fact 2 equations and that there are infinite solutions. And so to show the solution to the question, I display it with matrices (don't know how to use LaTeX for this but it's something like):
[x|  [1]   [4]
[y|=[2]+[5]t
[z|  [3]   [6]
Should I say or can it be anything? If so, how do I denote that? :P

Thanks

squance:
hmm...I'm not quite sure what you are asking here:
I have an example in my workbook that might help you:

Solve the linear system in the variables x, y ,z which has the row echelon form:

[1 0 1 -2
 0 2 2 4
 0 0 0 0]

Last row reads 0 = 0. z is not specified, so let z =t.

Second equation reads
2y + 2z = 4 ---> y + z =2
But z=t, so y = 2-t

First equation reads x + z = -2
but z=t, so x = -2-t

Hence solution is {(x,y,z): x=2-t, y =2-t,z=t, t for all real numbers}
or
{(x,y,z) = (2, 2, 0) + t(-1,-1,1), t for all real numbers}

Thats how I've taught to show the solution of a linear system.

ANd yes, in general you should say t for all real numbers...or if you have more than one parameter, eg: s and t, then you should write, "s,t for all real numbers"

Hope I've been some help

dekoyl:

--- Quote from: squance on February 07, 2009, 12:12:48 pm ---ANd yes, in general you should say t for all real numbers...or if you have more than one parameter, eg: s and t, then you should write, "s,t for all real numbers"

--- End quote ---
Thank you! Sorry I wasn't very clear but that was exactly what I was asking. ;D

And thanks for showing how you set out the solution. Instead of doing (x,y,z) we were shown in column form.

Oh and forgot to add, that Gaussian elimination method you posted in your thread was really helpful =] Thanks squance.

enwiabe:
Column/row form are both acceptable. Generally when giving a solution set, however, column form is the convention. Row form is the convention for describing a line in 3-space (generally).

dekoyl:
^Thanks enwiabe.

Another quick one:
When questions (related to matrices I did above) are like: "Find the values of k for which the system of equations has:
(i)one solution (ii)no solution" etc., when you work out the value of k that does satisfy the question, is the only way to show the solution by substituting into the matrix again to show that it satisfies the requirement?

Eg (my setting out for the answer):

If k = 3:

Infinitely many solutions

Edit: Sorry - I copy-pasted the LaTeX from my first post so the matrix didn't make sense

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