Linear: Find a set of basic solutions and show as linear combination

[sumtingwong59]

Homework Statement

Find a set of basic solutions and express the general solution as a linear combination of these basic solutions

a + 2b - c + 2d + e = 0
a + 2b + 2c + e = 0
2a + 4b - 2c + 3d + e = 0

Homework Equations

3. The Attempt at a Solution [/B]
i reduced it to:
1 2 0 0 -1 0
0 0 1 0 2/3 0
0 0 0 1 1 0

a = -2s + t
c = -2/3t
d = -t

I'm just not sure how i find solutions now. It could be literally anything could it not?

 

[LCKurtz]

Assuming your arithmetic is correct (I didn't check), you have let the free variables $b = s$ and $e = t$. I'm going to leave them as $b$ and $e$. Write your solution as$$
\left (\begin{array}{c}
a\\b\\c\\d\\e
\end{array}\right)
=
\left (\begin{array}{c}
-2b+e\\b\\-\frac 2 3 e\\-e\\e
\end{array}\right)
=
b\left (\begin{array}{c}
?\\?\\?\\?\\?
\end{array}\right)
+ e
\left (\begin{array}{c}
?\\?\\?\\?\\?
\end{array}\right)
$$Fill in the ?'s and you will have it.

 

[sumtingwong59]

Assuming your arithmetic is correct (I didn't check), you have let the free variables $b = s$ and $e = t$. I'm going to leave them as $b$ and $e$. Write your solution as$$
\left (\begin{array}{c}
a\\b\\c\\d\\e
\end{array}\right)
=
\left (\begin{array}{c}
-2b+e\\b\\-\frac 2 3 e\\-e\\e
\end{array}\right)
=
b\left (\begin{array}{c}
?\\?\\?\\?\\?
\end{array}\right)
+ c
\left (\begin{array}{c}
?\\?\\?\\?\\?
\end{array}\right)
$$Fill in the ?'s and you will have it.

Do I just pick any number to plug into b and e, and then pick different numbers and plug them into the c bracket?

 

[sumtingwong59]

Do I just want to make it so each equation equals 0?

 

[LCKurtz]

Do I just pick any number to plug into b and e, and then pick different numbers and plug them into the c bracket?

The $c$ in front of the last bracket was a typo; I have corrected it to $e$. Fill in the ?'s and there is nothing left to do. Every solution can be gotten for some value of $b$ and $e$. That is the general solution.