Class notes
homogenous system
- Course
- Institution
helps in understanding homogenous equations
[Show more]
Seller
Follow
oluwatosinolanrewaju
Content preview
University of Regina MATH 122 - Linear Algebra IMATH 122 - Linear Algebra I
§2.2. Associated homogeneous system
Martin Frankland
September 26, 2022
Example 1 (#2.2.5(c)). Express every solution of the system as a sum of a specific solution
plus a solution of the associated homogeneous system:
x1 + x2 − x3 − 5x5 = 2
x2 + x3 − 4x5 = −1
x2 + x3 + x4 − x5 = −1
2x1 − 4x3 + x4 + x5
= 6.
Solution. Let us write the augmented matrix of the system A⃗x = ⃗b and apply Gaussian
elimination:
1 1 −1 0 −5 2 1 1 −1 0 −5 2
h i 0 1 1 0 −4 −1 R −2R 0 1 1 0 −4 −1 −R2
R3∼
A ⃗b =
4∼ 1
0 1 1 1 −1 −1 0 1 1 1 −1 −1 R4 +2R2
2 0 −4 1 1 6 0 −2 −2 1 11 2
1 1 −1 0 −5 2 1 1 −1 0 −5 2 1 0 −2 0 −1 3
0 1 1 0 −4 −1 R4 −R3 0
1 1 0 −4 −1 R1 −R2 0
1 1 0 −4 −1
0 0 ∼ ∼ .
0 1 3 0 0 0 0 1 3 0 0 0 0 1 3 0
0 0 0 1 3 0 0 0 0 0 0 0 0 0 0 0 0 0
We use the free variables x3 and x5 as parameters, setting:
x3 = s, x5 = t.
The three equations yield
x1 = 3 + 2x3 + x5
= 3 + 2s + t
x2 = −1 − x3 + 4x5
= −1 − s + 4t
x4 = −3x5
= −3t
© 2022 Martin Frankland All Rights Reserved 1
, University of Regina MATH 122 - Linear Algebra I
so that the general solution is
x1 3 + 2s + t 3 2 1
x2 −1 − s + 4t −1 −1 4
x3 = s
⃗x = = 0 + s 1 + t 0
x4 −3t 0 0 −3
x5 t 0 0 1
for any s, t ∈ R. Here
3
−1
0
⃗xp =
0
0
is a particular solution and
2 1
−1 4
1 + t 0
⃗xh = s
0 −3
0 1
is the general solution of the associated homogeneous system A⃗x = ⃗0.
© 2022 Martin Frankland All Rights Reserved 2
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller oluwatosinolanrewaju. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $7.99. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
75759 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy study notes for 14 years now