Oxford Linear Algebra: Dimension Formula for Vector Spaces

Links to the other videos mentioned:

You can also find several video lectures from ProPrep explaining the content covered in the video at the links below.

As with all modules on ProPrep, each set of videos contains lectures, worked examples and full solutions to all exercises.

Watch the other videos from the Oxford Linear Algebra series at the links below.

The video begins by defining the dimension of a vector space as the number of elements in its basis. This is then exemplified by looking at the vector space of polynomials up to degree n, which has a dimension of n+1.

We then go through a fully worked example in R^4 by calculating explicitly the dimensions of the subspaces X and Y, the subspace X+Y, and the intersection of X and Y. This is used as motivation for the dimension formula: dim(X+Y) = dim(X) + dim(Y) - dim(X and Y).

Finally, a complete proof of the dimension formula is presented where we construct a basis of the space X+Y which is shown to be both spanning and linearly independent.

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[TomDarke-tojj]

Hi Tom

Earlier this year, I applied for physics at Merton and was rejected. During this time I had quite bad mental health problems and my grades and tests suffered. I know I scored reasonably well (just above average) on the entrance exam, but I came out feeling disappointed, knowing that I could have done better. I was unfortunately part of the minority that was not interviewed for my score.

Recently, I’ve been wondering whether to reapply. I am finally feeling more like myself and I’m quite happy because I’ve done really well on my mocks. If I were to reapply, would I have a reasonable shot of getting in? I’ve tried to search for the admissions statistics for re-applicants but I haven’t found anything. Also I know gap years are not favoured in STEM. Any advice would be amazing.

Also just want to say I love the channel. You always have really clear and well presented content and it’s much more Mathy than other channels. Please keep up the great work :).

[forthrightgambitia]

I must say I do love the smell of linear algebra in the morning.

[mijaelmarcelovillarroelchu]

Hello, I am a last year high school student in Bolivia, I am 17 years old, I decided to study mathematics permanently, but blindly because I don't know how much I like mathematics, or if I have an abstract mind for research, but still my love for math predominated despite the fact that I had already been deciding to study medicine but it was only because lately I have been losing myself in medical series and I had the illusion, in short, what depressed me about being a mathematician or even a physicist is that both sciences I feel that it has lost its essence, that is, it has been prostituted as an instrument of dissemination in order to make physics seem like child's play and mathematics like a spectacular adventure, in addition to the fact that many of their famous characters from both are so acquaintances that make you see the lives of these characters more than their work and even I have been very sympathetic with them since I value myself so little that I prefer to compare myself with people who had the same personality as me and triumph in life, so I sold myself that illusion. Another thing is that many famous mathematicians or physicists were born intelligent and gifted or else they became obsessed with matter from childhood and they investigated on their own and saw that they have talent, I have none of that, I chose study the degree as an epiphany and because if I was interested in those subjects, but I can't find a solid relationship with mathematics or physics, I can't and my worst fear is not knowing my talent or what I am suitable for, I perceive myself as someone insignificant who sometimes I think about death or the past, and as having been born in a country like Bolivia and being a student in a Catholic school in which I am an average student, even though I was recognized as the 3rd best, I am not able to assess This honor because it is from a school that I do not consider intellectually demanding or stimulating, I tried to join Cheenta's group (Indian youtube song about mathematics) but they rejected me or rather they eliminated me because at that moment I had given up being a mathematician, Now I'm looking for a mentor or tutor to help me strengthen my math skills. I hope someone cares to read this comment.

[sofilove...]

U are absolutely good teacher dude...With gorgeous style...

[israelruiz]

😍 I like Algebra.❤ It remembered me: Kerf (X), Img (X), canonical or orthonormal bases of X as a vectorial subspace of V.

[vardaanluthra]

Make a video on JEE advanced have mine in a one and a half month...
really enjoy watching ur videos

📔📕📖📗📙📘📚📓📑

[Ameslan]

Hey Dr Tom! Did you see the news that the Ukrainian high school girls mathematics team won the European Girls' Mathematical Olympiad that was just held in Slovenia? Congratulations to them!

[Jack-cmch]

Could you please do a series on what makes a good mathematician? Such things like what you taught Mike Boyd! Thank you kind sir

[Incognitooo]

Come-on, Tom, I am still waiting for an armwrestling between you and at least one of your maths students to be youtubed here !

[nolansu-hackett]

Tom could you do a video of you doing an Ib higher level aa math paper 1

[daxhausentraveller]

I have not got a clue about what your taking about . But I love watching maths 🧮 videos. 😊🤣

[belleringr]

Maybe Sir Isaac Newton rocks math too! 🤘prof. ever ⚽️🔢💯 #LOL