Uploaded March 2023 | Updated September 2026, 2 weeks ago
University of Oxford mathematician Dr Tom Crawford explains the terms basis, spanning and linear independence in the context of vectors and vector spaces. Check out ProPrep with a 30-day free trial to see how it can help you to improve your performance in STEM-based subjects: proprep.uk/info/TOM-Crawford
Test your understanding of the content covered in the video with some practice exercises courtesy of ProPrep. You can download the workbooks and solutions for free here: api.proprep.com/course/downloadbook?file=Proprep%20-%20Linear%20Algebra%20-%20Vector%20Spaces%20over%20R%20(Rn)%20-%20workbook.pdf
And here: api.proprep.com/course/downloadbook?file=Proprep%20-%20Linear%20Algebra%20-%20General%20Vector%20Spaces%20-%20workbook.pdf
You can also find several video lectures from ProPrep explaining the content covered in the video at the links below.
Basis for R^n: proprep.com/courses/all/linear-algebra/vector-spaces-over-r-(rn)/basis-for-rn/vid9962
Basis: proprep.com/courses/all/linear-algebra/general-vector-spaces/vector-basis/vid25730
Spanning: proprep.com/courses/all/linear-algebra/general-vector-spaces/linear-combination,-dependence-and-span/vid25722
Linear Independence: proprep.com/courses/all/linear-algebra/vector-spaces-over-r-(rn)/linear-combination,-dependence-and-span/vid10012
As with all modules on ProPrep, each set of videos contains lectures, worked examples and full solutions to all exercises.
Watch the other videos from the Oxford Linear Algebra series at the links below.
Solving Systems of Linear Equations using Elementary Row Operations (ERO’s): youtu.be/9pF__coVyEE
Calculating the inverse of 2x2, 3x3 and 4x4 matrices: youtu.be/VKOaG3Ogf9Q
What is the Determinant Function: youtube.com/watch?v=bLsBWVYSg0A
The Easiest Method to Calculate Determinants: youtu.be/qniUv4EZB0w
Eigenvalues and Eigenvectors Explained: youtu.be/8uISh6xyW7w
Spectral Theorem Proof: youtu.be/ADwsk9G5s_8
Vector Space Axioms: youtu.be/draqOOUoWQM
Subspace Test: youtu.be/3_MxBlWQsgs
The video begins with an intuitive example of a basis via the vector space of polynomials up to degree n. We then give the formal definition of a basis as a spanning set of linearly independent vectors.
The terms spanning and linear independence are then formally defined with examples given for each. We also show the definition of linear independence is equivalent to showing that the only solution to a linear combination of the vectors being equal to zero is for all of the coefficients to be zero. Linear dependence is defined as the lack of linear independence, or when a vector in a set can be written as a linear combination of the other vectors in the set.
Finally, we move on to a series of worked examples, beginning with several possible bases for the Cartesian plane R^2. We then look at examples of linearly independent and linearly dependent sets of vectors, and how to show this is the case. Finally, we construct two possible bases for 3D space R^3.
Produced by Dr Tom Crawford at the University of Oxford. Tom is an Early-Career Teaching and Outreach Fellow at St Edmund Hall: seh.ox.ac.uk/people/tom-crawford
For more maths content check out Tom's website tomrocksmaths.com
You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths.
facebook.com/tomrocksmaths
twitter.com/tomrocksmaths
instagram.com/tomrocksmaths
Get your Tom Rocks Maths merchandise here:
beautifulequations.net/collections/tom-rocks-maths
University of Oxford mathematician Dr Tom Crawford explains the terms basis, spanning and linear independence in the context of vectors and vector spaces. Check out ProPrep with a 30-day free trial to see how it can help you to improve your performance in STEM-based subjects: proprep.uk/info/TOM-Crawford
Test your understanding of the content covered in the video with some practice exercises courtesy of ProPrep. You can download the workbooks and solutions for free here: api.proprep.com/course/downloadbook?file=Proprep%20-%20Linear%20Algebra%20-%20Vector%20Spaces%20over%20R%20(Rn)%20-%20workbook.pdf
And here: api.proprep.com/course/downloadbook?file=Proprep%20-%20Linear%20Algebra%20-%20General%20Vector%20Spaces%20-%20workbook.pdf
You can also find several video lectures from ProPrep explaining the content covered in the video at the links below.
Basis for R^n: proprep.com/courses/all/linear-algebra/vector-spaces-over-r-(rn)/basis-for-rn/vid9962
Basis: proprep.com/courses/all/linear-algebra/general-vector-spaces/vector-basis/vid25730
Spanning: proprep.com/courses/all/linear-algebra/general-vector-spaces/linear-combination,-dependence-and-span/vid25722
Linear Independence: proprep.com/courses/all/linear-algebra/vector-spaces-over-r-(rn)/linear-combination,-dependence-and-span/vid10012
As with all modules on ProPrep, each set of videos contains lectures, worked examples and full solutions to all exercises.
Watch the other videos from the Oxford Linear Algebra series at the links below.
Solving Systems of Linear Equations using Elementary Row Operations (ERO’s): youtu.be/9pF__coVyEE
Calculating the inverse of 2x2, 3x3 and 4x4 matrices: youtu.be/VKOaG3Ogf9Q
What is the Determinant Function: youtube.com/watch?v=bLsBWVYSg0A
The Easiest Method to Calculate Determinants: youtu.be/qniUv4EZB0w
Eigenvalues and Eigenvectors Explained: youtu.be/8uISh6xyW7w
Spectral Theorem Proof: youtu.be/ADwsk9G5s_8
Vector Space Axioms: youtu.be/draqOOUoWQM
Subspace Test: youtu.be/3_MxBlWQsgs
The video begins with an intuitive example of a basis via the vector space of polynomials up to degree n. We then give the formal definition of a basis as a spanning set of linearly independent vectors.
The terms spanning and linear independence are then formally defined with examples given for each. We also show the definition of linear independence is equivalent to showing that the only solution to a linear combination of the vectors being equal to zero is for all of the coefficients to be zero. Linear dependence is defined as the lack of linear independence, or when a vector in a set can be written as a linear combination of the other vectors in the set.
Finally, we move on to a series of worked examples, beginning with several possible bases for the Cartesian plane R^2. We then look at examples of linearly independent and linearly dependent sets of vectors, and how to show this is the case. Finally, we construct two possible bases for 3D space R^3.
Produced by Dr Tom Crawford at the University of Oxford. Tom is an Early-Career Teaching and Outreach Fellow at St Edmund Hall: seh.ox.ac.uk/people/tom-crawford
For more maths content check out Tom's website tomrocksmaths.com
You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths.
facebook.com/tomrocksmaths
twitter.com/tomrocksmaths
instagram.com/tomrocksmaths
Get your Tom Rocks Maths merchandise here:
beautifulequations.net/collections/tom-rocks-maths
