Overview
On this page, we discuss linear independence of a set of vectors, as well as the concepts of basis, coordinates, and dimension.
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.
- Determine whether a set of vectors is linearly independent or linearly dependent.
- Express one vector in a linearly dependent set as a linear combination of the other vectors in the set.
- Show that a set of vectors is a basis for a vector space.
- Determine the dimension of a vector space
Advanced learning objectives
In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:
- Prove results about linear dependence and independence.
- Find the coordinates of a vector relative to a basis.
- Find the coordinate vector of a vector relative to a basis.
To prepare for class
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Watch this video (again) by 3Blue1Brown which explains the concepts of linear combinations, span, linear (in)dependence, and basis in \(\mathbb{R}^n\). Having a good understanding of this is crucial before moving on towards applying these concepts in general vector spaces.
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Watch this video by Lorenzo Sadun which gives the more technical definitions of linear independence and bases:
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Watch this video which explains how using a basis and coordinates with respect to that basis enables us to “identity” a given vector space (e.g. the vector space of polynomials of degree 2 or less) with an appropriate \(\mathbb{R}^n\) (\(n=3\) in this case) - and then use “usual vectors” in this \(\mathbb{R}^n\) to answer whatever question we were wondering about:
