Uploaded October 2023 | Updated September 2026, 2 weeks ago
University of Oxford Mathematician Dr Tom Crawford introduces the steps of the Gram-Schmidt Process and explains why the algorithm gives you an orthonormal set of vectors. Check out ProPrep with a 30-day free trial: proprep.uk/info/TOM-Crawford
Links to the other videos mentioned:
Inner Product Space youtube.com/watch?v=ywjA25xFuew
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-%20Inner%20Product%20Spaces%20-%20workbook.pdf
You can also find several video lectures from ProPrep explaining the content covered in the video here: proprep.com/courses/all/linear-algebra/inner-product-spaces/gram-schmitt-process/vid26063
As with all modules on ProPrep, each set of videos contains lectures, worked examples and full solutions to all exercises.
The video begins with a reminder of the definition of an orthonormal set, before introducing the 3 steps of the Gram-Schmidt Process. Step 1: normalise the first vector from a linearly independent set. Step 2: subtract the projection of the first orthonormal vector from the second vector in the linearly independent set, then normalise. Step 3: repeat step 2 for each of the remaining vectors.
Step 2 is explored in more detail through a direct calculation of the inner product and an explicit example in the 2D plane, including a visualisation of the projection map.
The video ends with a fully worked example of computing an orthonormal set in the polynomial inner product space where the inner product is defined via an integral.
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
Direct Sum: youtube.com/watch?v=b1VZm4Oecq4
Linear Transformations: youtu.be/UF59Mok4fsQ
Rank Nullity Theorem: youtu.be/yZ52iqxaiCc
Inner Product Space: youtube.com/watch?v=ywjA25xFuew
Produced by Dr Tom Crawford at the University of Oxford. Tom is Public Engagement Lead at the Oxford University Department of Continuing Education: conted.ox.ac.uk
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
Check out Proprep with a 30-day free trial here: proprep.uk/info/TOM-Crawford
University of Oxford Mathematician Dr Tom Crawford introduces the steps of the Gram-Schmidt Process and explains why the algorithm gives you an orthonormal set of vectors. Check out ProPrep with a 30-day free trial: proprep.uk/info/TOM-Crawford
Links to the other videos mentioned:
Inner Product Space youtube.com/watch?v=ywjA25xFuew
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-%20Inner%20Product%20Spaces%20-%20workbook.pdf
You can also find several video lectures from ProPrep explaining the content covered in the video here: proprep.com/courses/all/linear-algebra/inner-product-spaces/gram-schmitt-process/vid26063
As with all modules on ProPrep, each set of videos contains lectures, worked examples and full solutions to all exercises.
The video begins with a reminder of the definition of an orthonormal set, before introducing the 3 steps of the Gram-Schmidt Process. Step 1: normalise the first vector from a linearly independent set. Step 2: subtract the projection of the first orthonormal vector from the second vector in the linearly independent set, then normalise. Step 3: repeat step 2 for each of the remaining vectors.
Step 2 is explored in more detail through a direct calculation of the inner product and an explicit example in the 2D plane, including a visualisation of the projection map.
The video ends with a fully worked example of computing an orthonormal set in the polynomial inner product space where the inner product is defined via an integral.
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
Direct Sum: youtube.com/watch?v=b1VZm4Oecq4
Linear Transformations: youtu.be/UF59Mok4fsQ
Rank Nullity Theorem: youtu.be/yZ52iqxaiCc
Inner Product Space: youtube.com/watch?v=ywjA25xFuew
Produced by Dr Tom Crawford at the University of Oxford. Tom is Public Engagement Lead at the Oxford University Department of Continuing Education: conted.ox.ac.uk
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
Check out Proprep with a 30-day free trial here: proprep.uk/info/TOM-Crawford
