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: https://www.proprep.uk/info/TOM-Crawford Links to the other videos mentioned: Inner Product Space https://www.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: https://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: https://www.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): https://www.youtube.com/watch?v=9pF__coVyEE Calculating the inverse of 2x2, 3x3 and 4x4 matrices: https://www.youtube.com/watch?v=VKOaG3Ogf9Q What is the Determinant Function: https://www.youtube.com/watch?v=bLsBWVYSg0A The Easiest Method to Calculate Determinants: https://www.youtube.com/watch?v=qniUv4EZB0w Eigenvalues and Eigenvectors Explained: https://www.youtube.com/watch?v=8uISh6xyW7w Spectral Theorem Proof: https://www.youtube.com/watch?v=ADwsk9G5s_8 Vector Space Axioms: https://www.youtube.com/watch?v=draqOOUoWQM Subspace Test: https://www.youtube.com/watch?v=3_MxBlWQsgs Basis, Spanning and Linear Independence: https://www.youtube.com/watch?v=wdOFi8aUNp0 Dimension Formula: https://www.youtube.com/watch?v=lGo9E5dZ4f8 Direct Sum: https://www.youtube.com/watch?v=b1VZm4Oecq4 Linear Transformations: https://www.youtube.com/watch?v=UF59Mok4fsQ Rank Nullity Theorem: https://www.youtube.com/watch?v=yZ52iqxaiCc Inner Product Space: https://www.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: https://www.conted.ox.ac.uk/ For more maths content check out Tom's website https://tomrocksmaths.com/ You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths. https://www.facebook.com/tomrocksmaths/ https://twitter.com/tomrocksmaths https://www.instagram.com/tomrocksmaths/ Get your Tom Rocks Maths merchandise here: https://beautifulequations.net/collections/tom-rocks-maths Check out Proprep with a 30-day free trial here: https://www.proprep.uk/info/TOM-Crawford