Oxford Linear Algebra: Gram-Schmidt Process

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    • Oxford Linear Algebra: Inner Product ...  

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/downlo...

You can also find several video lectures from ProPrep explaining the content covered in the video here: https://www.proprep.com/courses/all/l...

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):    • Oxford Linear Algebra: Elementary Row...  

Calculating the inverse of 2x2, 3x3 and 4x4 matrices:    • Oxford Linear Algebra: How to find a ...  

What is the Determinant Function:    • Oxford Linear Algebra: What is the De...  

The Easiest Method to Calculate Determinants:    • Oxford Linear Algebra: The Easiest Me...  

Eigenvalues and Eigenvectors Explained:    • Oxford Linear Algebra: Eigenvalues an...  

Spectral Theorem Proof:    • Oxford Linear Algebra: Spectral Theor...  

Vector Space Axioms:    • Oxford Linear Algebra: What is a Vect...  

Subspace Test:    • Oxford Linear Algebra: Subspace Test  

Basis, Spanning and Linear Independence:    • Oxford Linear Algebra: Basis, Spannin...  

Dimension Formula:    • Oxford Linear Algebra: Dimension Form...  

Direct Sum:    • Oxford Linear Algebra: Direct Sum of ...  

Linear Transformations:    • Oxford Linear Algebra: Linear Transfo...  

Rank Nullity Theorem:    • Oxford Linear Algebra: Rank Nullity T...  

Inner Product Space:    • Oxford Linear Algebra: Inner Product ...  

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/

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