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 here.

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.