Overview
On this page, we discuss the geometry of the set of solutions of a linear system.
Important
The basic and advanced learning objectives listed below are meant to give you an idea of the material you should learn about this section. These are mainly intended to be used in a course which uses an Active Learning approach, where students are required to “read ahead” before each class - but can equally be used in a more traditional course setting.
Unless your teacher gives you specific instructions, it is up to you to decide how much of the listed resources you need to read or watch - you probably do not need to go through all of it. You might also want to look at the General Study Tips & Tricks page for some recommendations on how to effectively study with a math textbook and videos.
Basic learning objectives
These are the tasks you should be able to perform with reasonable fluency when you arrive at your next class meeting. Important new vocabulary words are indicated in italics.
- Understand that every homogeneous system of linear equations always has at least the trivial solution \(\vec{x}=\vec{0}\).
- Determine by inspection from the RREF of the coefficient matrix if a homogeneous system has non-trivial solutions.
Advanced learning objectives
In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:
- Use a specific solution to the nonhomogeneous linear system \(A\vec{x}=\vec{b}\) and the general solution of the corresponding linear system \(A\vec{x}=\vec{0}\) to obtain the general solution to \(A\vec{x}=\vec{b}\).
To prepare for class
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Watch this video which shows that a homogeneous system of linear equations can only have non-trivial solutions if there is at least one free variable:
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Watch this video which explains the relationship between the set of solutions of a (non-homogeneous) SLE and the set of solutions of the corresponding homogeneous system:
