Oxford Linear Algebra: Gram-Schmidt Process
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- เผยแพร่เมื่อ 21 ต.ค. 2023
- 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: www.proprep.uk/info/TOM-Crawford
Links to the other videos mentioned:
Inner Product Space • Oxford Linear Algebra:...
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/downlo...
You can also find several video lectures from ProPrep explaining the content covered in the video here: 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:...
Calculating the inverse of 2x2, 3x3 and 4x4 matrices: • Oxford Linear Algebra:...
What is the Determinant Function: • Oxford Linear Algebra:...
The Easiest Method to Calculate Determinants: • Oxford Linear Algebra:...
Eigenvalues and Eigenvectors Explained: • Oxford Linear Algebra:...
Spectral Theorem Proof: • Oxford Linear Algebra:...
Vector Space Axioms: • Oxford Linear Algebra:...
Subspace Test: • Oxford Linear Algebra:...
Basis, Spanning and Linear Independence: • Oxford Linear Algebra:...
Dimension Formula: • Oxford Linear Algebra:...
Direct Sum: • Oxford Linear Algebra:...
Linear Transformations: • Oxford Linear Algebra:...
Rank Nullity Theorem: • Oxford Linear Algebra:...
Inner Product Space: • Oxford Linear Algebra:...
Produced by Dr Tom Crawford at the University of Oxford. Tom is Public Engagement Lead at the Oxford University Department of Continuing Education: www.conted.ox.ac.uk/
For more maths content check out Tom's website tomrocksmaths.com/
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Check out Proprep with a 30-day free trial here: www.proprep.uk/info/TOM-Crawford
Check out ProPrep with a 30-day free trial: www.proprep.uk/info/TOM-Crawford
The 30-day trial doesn't work
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!
We just covered this in my undergraduate linear algebra class!
Great work professor!
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)
Thanks for bringing higher math to the "masses" 🙂
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
At 16:21 is there a difference between the x written like 2 c's and the x written with straight lines?
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
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
I think it's just a way to define the inner product in that question. It doesn't break any of the properties needed for it to be an inner product, so, it's valid.
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?
🐐🔥
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!
Nah…. You could do it. You just have to work up the this point.
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.
Orthonormal basis yes
Please do modified graham schmidt
Hi tom
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
What's Your Height ?
Yo I have that shirt
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
Do you have any maths videos for normal people, who are not oxbridge educated?
Dude this is for normal people, it’s just linear algebra