Oxford Linear Algebra: Subspace Test

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.

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.

[phenixorbitall]

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!

[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!

[christianmiranda]

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

[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?

[VolumetricTerrain-hzci]

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!

[jacksonmadison]

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

[michter]

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

[zizou]

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?

[eswyatt]

No. Vectors in R2 only have 2 entries!!

[unnamedchannelowouwu]

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