Uploaded May 2023 | Updated September 2026, 2 weeks ago
University of Oxford mathematician Dr Tom Crawford explains the concept of the direct sum of vector subspaces with several fully-worked examples. 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
Links to the other videos mentioned:
Subspace Test - youtu.be/3_MxBlWQsgs
Dimension Formula: youtu.be/lGo9E5dZ4f8
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-%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.
Direct Sum: proprep.com/courses/all/linear-algebra/general-vector-spaces/subspaces/vid30219
Sum of Subspaces: proprep.com/university-of-warwick/warwick-university/ma106/general-vector-spaces/subspaces/vid30220
As with all modules on ProPrep, each set of videos contains lectures, worked examples and full solutions to all exercises.
The video begins with the formal mathematical definition of a direct sum via the sum of vector spaces which only have the zero vector in their intersection. This is then shown to be equivalent to an alternative definition which says the representation via the sum of vectors is unique.
Next we look at 3 fully-worked examples where we are asked to check if the given sum of vector spaces is a true direct sum or not. By checking the intersection we see that two of them are not a direct sum as their intersection contains a non-zero vector. For the third example the intersection is shown to be zero, so we then check whether the two subspaces span the larger vector space. By constructing a basis for each subspace, we see that this is indeed the case and thus conclude it is a true direct sum.
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
Basis, Spanning and Linear Independence: youtu.be/wdOFi8aUNp0
Dimension Formula: youtu.be/lGo9E5dZ4f8
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 concept of the direct sum of vector subspaces with several fully-worked examples. 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
Links to the other videos mentioned:
Subspace Test - youtu.be/3_MxBlWQsgs
Dimension Formula: youtu.be/lGo9E5dZ4f8
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-%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.
Direct Sum: proprep.com/courses/all/linear-algebra/general-vector-spaces/subspaces/vid30219
Sum of Subspaces: proprep.com/university-of-warwick/warwick-university/ma106/general-vector-spaces/subspaces/vid30220
As with all modules on ProPrep, each set of videos contains lectures, worked examples and full solutions to all exercises.
The video begins with the formal mathematical definition of a direct sum via the sum of vector spaces which only have the zero vector in their intersection. This is then shown to be equivalent to an alternative definition which says the representation via the sum of vectors is unique.
Next we look at 3 fully-worked examples where we are asked to check if the given sum of vector spaces is a true direct sum or not. By checking the intersection we see that two of them are not a direct sum as their intersection contains a non-zero vector. For the third example the intersection is shown to be zero, so we then check whether the two subspaces span the larger vector space. By constructing a basis for each subspace, we see that this is indeed the case and thus conclude it is a true direct sum.
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
Basis, Spanning and Linear Independence: youtu.be/wdOFi8aUNp0
Dimension Formula: youtu.be/lGo9E5dZ4f8
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
