Oxford Linear Algebra: Gram-Schmidt Process

Links to the other videos mentioned:

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.

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[franciscodanieldiazgonzale]

Congratulations for your appointment as Public Engagement Lead and a Departmental Lecturer at the department of continuing education in University of Oxford. We have an Oxford Mathematics Lecturer here people!

[stevenbercik]

We just covered this in my undergraduate linear algebra class!

[Shaan_Suri]

In the last example, the two vectors only vectors are a basis of polynomials with degree less than or equal to 2. But let's say our vector space was polynomials with degree less than or equal to 3, and we were only given 2 linearly independent vectors, is there a way to construct an orthonormal basis? (ie. extend the orthonormal set with 2 vectors which you found into a set with 3 vectors)

[bread]

At 16:21 is there a difference between the x written like 2 c's and the x written with straight lines?

[walterblair]

Hey I reckon you should take a look at the Australian HSC math exams. There's 3 different "hard" ones. Advanced, ext 1 and ext 2. You may just have to brush up on the calculator

[eugenek]

Thanks for bringing higher math to the "masses" 🙂

[narfwhals]

Here you show that, given an inner product and a linear independent set we can always create an orthonormal set. Can we go the other way? Can we take a linearly independent set, set it to be orthonormal, and always find an inner product that satisfies this?

[jorgesalazar]

What are the reasons for take\int xf(x)g(x)dx as inner product in your last example? Is possible to do G-S process fir a two dimensional space? thanks for your videos

[timothymattnew]

Isn't Gram-Schmidt process that somehow very intuitively understandable process of turning any basis of an Euclidean space into an ortnonormal basis? I never thought it even had a name because it seems so straightforward.

[michelesereni]

hello, im sorry if i dont grasp it yet, but isnt the example in the polynomial space not an inner space? shoulnt there be absolute values? if i take g(x) = -x, f(x) = x, then i have the integral of -x^3, which amounts to a negative value? Am I right? or what am I missing? Thank you for your help

[pranavkarthik]

I'm far too dumb to watch this or else i would ... Maybe if i don't have headache or my mother hounding me if i don't get off screen. Love you lots!

[somanandi]

Hi can you please react to jee advanced math paper. It is very tough. It is an exam given by highschool students to get into respectable colleges IIT

[mathswala]

Hlo, give answer of this question.
Number of tangents of curve y=e^|x| at (0, 1),
Options are
a) 2, b) 4, c) 1, d) 0

[Gj-]

Do you have any maths videos for normal people, who are not oxbridge educated?