Oxford Linear Algebra: Inner Product Space
ฝัง
- เผยแพร่เมื่อ 16 ก.ย. 2023
- University of Oxford Mathematician Dr Tom Crawford introduces the concept of a Bilinear Form, Inner Product, Sesquilinear Form and Inner Product Space. Check out ProPrep with a 30-day free trial: www.proprep.uk/info/TOM-Crawford
Links to the other videos mentioned:
Linear Transformations: • Oxford Linear Algebra:...
Spanning, Basis & Linear Independence: • 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 the definition of a Bilinear Form with a concrete example of the dot product on R^n. This is shown to also satisfy the criteria to be symmetric and positive definite, thus making it an Inner Product.
The concept of an Inner Product Space is then introduced as a Real Vector Space equipped with an Inner Product. A second example involving an integral over the space of real polynomials is then explored.
In the second part of the video Orthonormal Sets are introduced via a definition and then the proof of a lemma stating that any Orthonormal Set in an Inner Product Space is Linearly Independent. The video concludes with a final definition of a Sesquilinear Form and a discussion of a Complex Inner Product Space.
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:...
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/
You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths.
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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
why are we integrating from 0 to 1 while proving polynomial function space as a inner product space
It is strictly greater than 0 not greater than. Or equal to
I finally decided to get straight on some definitions; seems like I came to the right place. Well done!
Feedback: It would be really helpful if you could add (for those who require it), that an inner product requires conjugate symmetry. It just so happens that if you’re dealing with real fields then conjugate symmtery becomes your normal symmetry.
I believe these small sidenotes would help point people, who are looking for more advanced linear algebra concepts, in the right direction.
It would also make it easier for people to make connections between what’s presented here, and what is popularly found online. (i.e: Wikipedia)
This video was surprisingly easy to follow. Its been years since I have done anything involving Linear Algebra but I was able to follow the video thoroughly.
Oh, and nice video too. Very well explained.
Tomrocksmath you are really good at this!
Thanks for sharing
Hey, your content is just amazing. Here's a challenge: do the ITA 2023 math exam, that's one of the hardest Brazil tests for college acceptance.
this professor it's so cool, and btw i liked your tattoos. I hope you continue with your channel it's pertinant to any student or interested in Math.
What I don't like about inner product spaces is the name. I think important mathematical concepts ought to have short names. For example, vector spaces with a norm are called normed spaces. This is short and sweet. Inner products and their variants are so ubiquitous that they really ought to have a more concise and compact name.
Also, I think names should be explained. What is 'inner' about inner product spaces? And does this mean there is also an outer product in contrast? And if so, how do they relate?
I've seen certain specialisations of the tensor product called outer products. So I guess that answers one of my questions. And I expect its called 'inner' because it reduces the 'dimension' of the product - from a vector to a scalar. Whilst an 'outer' product increases the dimension, from a pair of vectors to a 2-vector.
What I'm driving at here is that maths has a history thats embedded in its notation amd naming strategy. And I think that its worth making explicit.
My suggestion for an alternative name for an inner product space is one thats imported from physics - a metric. Thus a metric space. Unfortunately this clashes with the standard notion of a metric space in maths. I guess we could distinguish the former from the latter by calling the former, a metric vector space and the latter, a topological metric space.
And my vote for the most awkward name in maths goes to sesquilinear form. This, is a bilinear form that is linear in one variable and conjugate linear in the other. Since this comes up all the time for vector spaces over the complex numbers, we ought to have a nicer name for this too.
was at open day on friday. shame didnt see you tom
Embarrassingly, I forgot the inner product and this video released at a perfect time for me to remember all about it!
I don't understand why that is "embarrassing".
Does it have "a" orthonormal basis, or does it have "at least one" orthonormal basis?
🔥 🔥 🔥
Can you do the 2023 higher leaving cert paper (irish equivalent of gcse/a levels)
Applied maths and maths
But from what I learnt, positive definite is strictly positive numbers excluding zero.
So please let's see and maybe make correction on it
Could someone tell me what the V x V notation means?
Vector X Vector
Oxford linear algebra. It’s like regular linear algebra, but Oxford
What came first, Maths or science?
Maths. Because maths can exist without science and even the number 1 can be classified as maths and started in the time of cavemen before people could talk
Neither, you came first
Since I am building a perpetual motion machine that Gottfried Leibniz watched run for 2 hours, when he published his work, was Newton correct when Newton said it was his own work? Did Leibniz learn of Newton's unpublished work which only Newton's friends knew of? Because I am 1/2 Norwegian I basically learned to talk in Norway. Bessler was 1/2 Polish and Leibniz was a witness for Bessler when Bessler was arrested for being a fraud. Bessler wasn't convicted. To be a jerk I'll show how Newton could've used his math to show why the Moon orbits the Earth. Johannes Keppler used the inverse square law to understand the motion of the planets around the Sun. If Newton would've been tolerant of Keppler then he would've been a genius. I just explained Newton's most famous work. I think everyone will miss what I just said. It's not like the letter that Newton wrote to Keppler but is much more direct. Nothing personal Tom but people need to understand that science and math like anything else involves mind games and politics.
Jai shree ram
Does water float in water or is it submerged?