Linear Algebra - Lecture 28 - Subspaces

In this lecture, we define subspaces and view some examples and non-examples.

[toxicwasteofspace]

You have no idea how grateful I am for these lectures. You are literally saving my uni career lol. Thank you so much.

[shazadqureshi]

thank you after watching a couple videos and not understanding anything, I came across this and I actually got it!

[abdullahalrafi]

Some can't give the education taking money what you are giving for free. Thanks a lot sir. Hope my exams will be great.

[jeremieniyonsaba]

Thank you sir, i really struggled to get an idea about subspaces but now you have helped a lot

[TheBearCoder]

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[maryjoytasong]

Thank You for saving me in my online class 😭😭💛💛💛💛💛💛

[gpvvowg]

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[suprememaster]

Every subspace of R5 that contains a nonzero vector must contain a line. Is this statement true?

[dktchr]

At ~7:13, am unclear how the "copy" of R2 can be a subspace since even if the vector has a 0 for one of its entries, it still has 3 entries, not two. This would seem to contradict the definition for closure.. Your thoughts?

[faisalsal]

How can we say that subspace "inherits" any property from its parent space?
If V is a space, then U (some elements of V) will qualify to be a "subspace" only if they fulfill all the 10 properties.
if V is a Vector space of R2
and U is a subset of V where both entries must be Even.
and W is a subset of V where both entries must be Odd.
Is U a sub-space?
Yes because adding two even numbers or multiplying with Scalar will still be an Even number. [4 4]+[2 4] = [6 8], likewise 5[4 6] = [20 30].
[6 8] and [20 30] belong to U as both its components are Even and that satisfies it to be an element of U. We pick two Vectors that belong to U and by doing any operation, the output remains in U. U is a "sub-space".
Is W is a subspace?
No, because adding two odd numbers or multiplying with an "even" Scalar will NOT result in odd numbers. [3 3]+[1 5] = [4 8], likewise 4[3 9] = [12 36]
[4 6] and [12 36] do NOT belong to W as neither of its components is Odd. So we start with two Vectors[3 3] and [1 5] that belong to W (as both its components are odd) but by doing "additions" and "multiplication by Scalar", the output is not W. So W is not "sub-space".