Which of the following sets of vectors span conjugated subspaces of ℂ^3 ? (a) ( 1 -1

step 1 Hello and welcome to problem 2 .1 .2. You were asked to show if these subspaces or subsets are s

Which of the following sets of vectors span conjugated...

Key Concepts

Complex Vector Spaces

A complex vector space is a set of vectors where the scalars come from the field of complex numbers. In these spaces, vectors can have complex entries and linear combinations are formed using complex coefficients. Understanding complex vector spaces is essential for discussing any topics related to spans, subspaces, or transformations such as complex conjugation.

Spanning Sets

A spanning set for a vector space or subspace is a set of vectors such that any vector in the space can be written as a linear combination of them. The concept of a span is central when determining whether a given set of vectors generates a particular subspace. This idea is used to decide if the provided vectors are sufficient to fully describe a subspace of ?^3.

Linear Independence

Linear independence is a property of a set of vectors that assures no vector in the set can be written as a linear combination of the others. When considering spanning sets, it is usually desirable (though not necessary) to have a basis where the vectors are linearly independent, meaning that the subspace has the minimum number of vectors needed to describe it.

Complex Conjugation

Complex conjugation is an operation on complex numbers (and by extension on the entries of vectors in a complex vector space) that replaces each entry with its complex conjugate. This operation is an involution, meaning that applying it twice returns the original number or vector, and it plays a key role in many areas including the study of conjugate subspaces.

Conjugate Subspaces

Given a subspace of a complex vector space, its conjugate subspace is obtained by taking the complex conjugate of every vector in the original subspace. Two subspaces are called conjugate if one is the image of the other under complex conjugation. Understanding this concept is crucial when determining if the vectors provided span subspaces that are related by conjugation in ?^3.

Transcript

00:01 Hello and welcome to problem 2 .1 .2.

00:03 You were asked to show if these subspaces or subsets are subspaces of r3.

00:09 And here it's really important to know what does that mean? what does a subspace mean? well, given a larger vector space, something is a subspace of that if it is closed under scalar multiplication and vector addition.

00:26 That's it.

00:27 So for this, we have, for part a, we're given the vector 0, b2, b3.

00:39 Okay.

00:40 And we're asked, well, is this a subspace of r3, given that b1 and b2 are real numbers? and the first part is to show if it's closed under vector addition.

01:00 So let's say we have 0, a1, a2, plus 0, b1, b2.

01:13 Well, when we add these together, we will have a zero on the top, a1 plus b1, and b2, or a, whatever, we'll do b2.

01:24 Plus a2.

01:26 And this is closed under addition because a1 plus b1 is just a value in r3 and obeys the same rules as as the original.

01:42 So yes, it's closed under vector addition.

01:45 What about scalar multiplication? well, by the same thing or by similar logic, we'll multiply a vector by constant c.

01:53 We'll have 0, a1, a2.

01:59 And when we multiply this, we'll have the vector 0, c, a1, c, a2.

02:08 And this is a vector in the subspace of r3 obeying the original rules.

02:14 So the answer to this part, part a, is yes.

02:20 And let's move on to part b.

02:25 Part b is the plane of vectors with b1 equals 1.

02:30 So let's do green for this one.

02:33 So just something with 1, something b2, and b3.

02:39 We're asked, is this a subspace of r3? well, we can solve this pretty quickly.

02:46 We see this is not closed under a vector addition.

02:50 If we add this with something else like 1a2, a3, we're going to get something with a 2 on top.

03:00 And that's not allowed.

03:01 We can only have something with ones on top.

03:03 So this is not a subspace.

03:07 It's good at part c.

03:11 Something with the union of two subspaces with b2 is zero and b3 is zero.

03:17 Okay, so how are we going to write this? let's write this as the subspace s, which is the union of a and b.

03:32 Now b is everything where let's see, b2 equals 0.

03:44 I'm not doing proper notation here.

03:47 I'm just trying to shorthand.

03:49 But b is the one where b3 equals 0.

03:53 So let's give an example of something in subset a.

04:01 Well, we'll have a vector a1, 0, a3.

04:10 Okay.

04:11 And what about in b? we'll have something that's b1, b2, 0.

04:22 All right, and s is just the union of that.

04:25 So we're asked, is s closed under vector addition? and what we can do is just add these two things together, and we can see that it does not, actually.

04:39 If we add these two together, we'll have a1 plus b1, we'll have b2, and a3.

04:51 And in this case, the second and the third values are, they don't multiply to zero...

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