Oxford Linear Algebra: Subspace Test

University of Oxford mathematician Dr Tom Crawford explains the subspace test for vector spaces. Check out ProPrep with a 30-day free trial to see how it can help you to improve your performance in STEM-based subjects: www.proprep.uk...
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: www.proprep.uk...
And here: www.proprep.uk...
You can also find several video lectures from ProPrep explaining subspaces here: www.proprep.uk...
And further videos explaining subspaces for more general vector spaces here: www.proprep.uk...
As with all modules on ProPrep, each set of videos contains lectures, worked examples and full solutions to all exercises.
Watch 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:...
The video begins with the definition of a subspace U contained in a vector space V, and some trivial examples for U = V and U = 0. The subspace test is then introduced and shown to be equivalent to the definition. The subspace test requires the zero vector to be contained in U, and any linear combination of vectors in U to also be contained in U. Finally, 3 fully worked examples are shown. First, we show that the x-y plane is a subspace of 3-dimensional coordinate space. Second, we show that for U and W subspaces of a vector space V, the intersection of U and W is always a subspace. Third, we show that the subspace of differentiable functions from the real numbers to the real numbers is a subspace of the vector space of all functions from R to R.
Produced by Dr Tom Crawford at the University of Oxford. Tom is an Early-Career Teaching and Outreach Fellow at St Edmund Hall: www.seh.ox.ac....
For more maths content check out Tom's website tomrocksmaths....
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Get your Tom Rocks Maths merchandise here: beautifulequat...

[@TomRocksMaths]

Check out ProPrep with a 30-day free trial to see how it can help you to improve your performance in STEM-based subjects: www.proprep.uk/info/TOM-Crawford

[@phenixorbitall3917]

I really like your last example!!! The abstract nature of mathematics is certainly its greatest strength, but it is also the reason why it is never quite clear to me at first, what given axioms - which, by the way, always seem to just fall out of the sky - and the theorems that follow from them have to do with reality and why they matter in life. Therefore I thank you very much for ending the video with something very tangible: Functions are by far the mathematical objects I am most familiar with! 🙂👍 I've always found linear algebra to be pretty dry, but your video makes me want to look at the subject with new eyes: your enthusiasm for math is really contagious and I like your structured way of explaining it! I'll try to check out ProPrep - Thank you very much for the hint!

[@christianmiranda331]

Fantastic professor! I love how he is able to communicate these concepts in a really digestible manner.

[@mrigayu]

Thanks for the clear and thorough explanation! I'm taking a Differential Equations and Linear Algebra class right now. We're about to introduce vector spaces in class, so I really appreciate this explanation!

[@alimumtazR]

You are amazing, professor! ❤

[@williamroywhite]

Hi Tom, thanks for another great lecture. I may be being a little picky but I thought that strictly speaking R2 is NOT a subspace of R3 since the vector (0,y,z), (x,0,z) and (x,y,0) are not equal and therefore should not be described as R2? In other words, the xy plane, xz plane and yz plane are not the same thing. Indeed, these are all examples of 2D planes embedded in R3 and actually there are infinitely many of them. So should we not be saying, in this example, that the subspace is the xy plane rather that R2?

[@xarlition]

Yes, you are thinking right. (0,y,z)(0,y,z), (x,0,z)(x,0,z) and (x,y,0)(x,y,0), each embedded in the R3R3 space. The vectors are actually instances of 2D planes that form subspaces of R3R3.Each of these vectors defines a plane, but these planes are in R3R3, not in R2R2. Therefore, these planes are subspaces of R3R3 space and are considered as subspaces of R3R3 space. Although each plane has the same size as plane R2R2, these planes are contained within R3R3, not within R2R2. Therefore, instead of defining these planes as R2R2, it would be more appropriate to think of them as R3R3 subspaces contained within R3R3. As a result, the xy plane, xz plane and yz plane are all subspaces of the R3R3 space, and therefore instead of defining the subspace as R2R2, it would be more accurate to consider these subspaces within R3R3.

[@VolumetricTerrain-hz7ci]

There are unknown way to visualize subspace, or vector spaces.
You can stretching the width of the x axis, for example, in the right line of a 3d stereo image, and also get depth, as shown below.
L R
|____| |______|
TIP: To get the 3d depth, close one eye and focus on either left or right line, and then open it.
This because the z axis uses x to get depth. Which means that you can get double depth to the image.... 4d depth??? :O
p.s
You're good teacher!

[@michter765]

I am ready to join the competition, where can I sign in?

[@TomRocksMaths]

seh.ac/teddyrocksmaths

[@zizou2076]

I have a linear Algebra mid-term exam next week can you please make a video about Spanning subsets/spanned subspaces or about basis and dimension?

[@TomRocksMaths]

Coming in 10 days or so if I get my editing done

[@jacksonmadison9994]

24:00
I thought R2 is not a sub space of R3 because they have different dimensions

[@jwandao3919]

I don’t get that part as well. have you understood it yet?

[@eswyatt]

Correct. The video is wrong

[@eswyatt]

No. Vectors in R2 only have 2 entries!!

[@unnamedchannelowouwu]

Can anyone from any part of the world sing the Comp?

[@TomRocksMaths]

Yes - anyone is eligible

[@derrickli497]

My mid term is on you man😂