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

  • #1

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?
 

Answers and Replies

  • #2
LCKurtz
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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.
 
Last edited:
  • #3
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 in to b and e, and then pick different numbers and plug them into the c bracket?
 
  • #4
Do I just want to make it so each equation equals 0?
 
  • #5
LCKurtz
Science Advisor
Homework Helper
Insights Author
Gold Member
9,519
734
Do I just pick any number to plug in to 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.
 
Last edited:

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