Uploaded February 2023 | Updated September 2026, 2 weeks ago
University of Oxford mathematician Dr Tom Crawford explains the subspace test for 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: proprep.uk/Academic/DownloadBook?file=Vector%20Spaces%20over%20R%20(Rn)%20-%20workbook%20uk.pdf
And here: proprep.uk/Academic/DownloadBook?file=Proprep%20-%20Linear%20Algebra%20-%20General%20Vector%20Spaces%20-%20workbook%20uk.pdf
You can also find several video lectures from ProPrep explaining subspaces here: proprep.uk/general-modules/all/linear-algebra/vector-spaces-over-r-(rn)/subspaces/vid9982
And further videos explaining subspaces for more general vector spaces here: proprep.uk/general-modules/all/linear-algebra/general-vector-spaces/subspaces?ucid=0
As with all modules on ProPrep, each set of videos contains lectures, worked examples and full solutions to all exercises.
Watch 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
The video begins with the definition of a subspace U contained in a vector space V, and some trivial examples for U = V and U = 0. The subspace test is then introduced and shown to be equivalent to the definition. The subspace test requires the zero vector to be contained in U, and any linear combination of vectors in U to also be contained in U. Finally, 3 fully worked examples are shown. First, we show that the x-y plane is a subspace of 3-dimensional coordinate space. Second, we show that for U and W subspaces of a vector space V, the intersection of U and W is always a subspace. Third, we show that the subspace of differentiable functions from the real numbers to the real numbers is a subspace of the vector space of all functions from R to R.
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 subspace test for 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: proprep.uk/Academic/DownloadBook?file=Vector%20Spaces%20over%20R%20(Rn)%20-%20workbook%20uk.pdf
And here: proprep.uk/Academic/DownloadBook?file=Proprep%20-%20Linear%20Algebra%20-%20General%20Vector%20Spaces%20-%20workbook%20uk.pdf
You can also find several video lectures from ProPrep explaining subspaces here: proprep.uk/general-modules/all/linear-algebra/vector-spaces-over-r-(rn)/subspaces/vid9982
And further videos explaining subspaces for more general vector spaces here: proprep.uk/general-modules/all/linear-algebra/general-vector-spaces/subspaces?ucid=0
As with all modules on ProPrep, each set of videos contains lectures, worked examples and full solutions to all exercises.
Watch 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
The video begins with the definition of a subspace U contained in a vector space V, and some trivial examples for U = V and U = 0. The subspace test is then introduced and shown to be equivalent to the definition. The subspace test requires the zero vector to be contained in U, and any linear combination of vectors in U to also be contained in U. Finally, 3 fully worked examples are shown. First, we show that the x-y plane is a subspace of 3-dimensional coordinate space. Second, we show that for U and W subspaces of a vector space V, the intersection of U and W is always a subspace. Third, we show that the subspace of differentiable functions from the real numbers to the real numbers is a subspace of the vector space of all functions from R to R.
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
