Overview
On this page, we discuss subspaces of a given vector space.
Note: 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.
 Determine whether a subset of \(\mathbb{R}^n\) is a subspace.
 Determine whether a subset of \(M_{mn}\) is a subspace.
 Show that a subset of \(\mathbb{R}^n\) or \(M_{mn}\) is a subspace.
 Show that a nonempty subset of \(\mathbb{R}^n\) or \(M_{mn}\) is not a subspace by demonstrating that the set is either not closed under addition or not closed under scalar multiplication.
 Given a set \(S\) of vectors in \(\mathbb{R}^n\) and a vector \(v\) in \(\mathbb{R}^n\), determine whether \(v\) is a linear combination of the vectors in \(S\).
 Given a set \(S\) of vectors in \(\mathbb{R}^n\), determine whether the vectors in \(S\) span \(\mathbb{R}^n\).
 Determine whether two nonempty sets of vectors in a vector space \(V\) space the same subspace of \(V\).
Advanced learning objectives
In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:
 Determine whether a subset of a vector space is a subspace.
 Show that a subset of a vector space is a subspace.
 Show that a nonempty subset of a vector space is not a subspace.
 Given a set \(S\) of vectors in a vector space \(V\) and a vector \(v\) in \(V\), determine whether \(v\) is a linear combination of the vectors in \(S\).
 Given a set \(S\) of vectors in a vector space \(V\), determine whether the vectors in \(S\) span \(V\).
 Determine whether two nonempty sets of vectors in a vector space \(V\) space the same subspace of \(V\).
To prepare for class

Watch this video (by James Hamblin) which defines the concept of a subspace, and goes through several important examples: