In Exercises 17 and 18, all vectors and subspaces are in ℝ^n . Mark each statement True or False

In Exercises 17 and 18, all vectors and subspaces are in ℝ^n...

Key Concepts

Column Space

The column space of a matrix is the set of all possible linear combinations of its column vectors. It corresponds to the subspace that these columns span and is pivotal in understanding solutions to linear systems, as it relates directly to the range of the linear transformation represented by the matrix.

Projection

Projection in a vector space is the process of mapping a vector onto a subspace such that the difference between the vector and its projection is orthogonal to the subspace. The projection operator is central in optimization and approximation methods, as it provides the closest representation of a vector within the subspace.

Orthogonality

Orthogonality refers to the relationship between vectors where their inner product is zero, meaning they are perpendicular to each other in inner product spaces. An orthogonal set simplifies many computations, such as projections and decompositions, and can be normalized to form an orthonormal basis.

QR Factorization

QR Factorization is a method of decomposing a matrix into the product of an orthogonal matrix Q and an upper triangular matrix R. This factorization is valuable in solving linear systems, least squares problems, and eigenvalue computations, as it helps reveal the underlying geometric structure of the matrix.

Basis

A basis is a set of linearly independent vectors that spans a vector space or subspace. It provides the minimal and unique (up to ordering and scaling in the case of inner product spaces) way of representing every vector in that space. The concept of a basis is fundamental in understanding dimensions and coordinate representations.

Span

Span refers to the set of all possible linear combinations of a given set of vectors. It represents the subspace that those vectors can generate within a vector space. Understanding span is crucial for determining whether a set of vectors can cover or represent an entire space or subspace.

Linear Independence

Linear independence is a property of a set of vectors where no vector in the set can be written as a linear combination of the others. This concept is essential for establishing a basis and ensuring that representations of vectors in a space are unique and non-redundant.

Transcript

00:01 So true or false, we're given that w is the span of x1, x2, x3, where x1, x2, x3 are linearly independent.

00:08 And then v1, v2, v3 is orthogonal in w.

00:12 Okay? does this mean that v1, v2, v3 is a basis for w? now, always make sure you check your definitions.

00:21 So let's go to page 340.

00:24 What is an orthogonal set? a set of vectors u1 up to up, they say.

00:30 In rn is orthogonal means that that each pair of distinct vectors are orthogonal.

00:39 So ui dot uj is equal to zero for every i not equal to j that's between one and p.

00:48 Okay, so please notice the very important fact that this doesn't rule out the case where all the vectors in my set are zero.

00:58 Okay, so two vectors could be orthogonal to each other, simply because at least one of the vectors are zero.

01:07 So this is where the meaning of orthogonality slightly departs from our intuition perhaps.

01:13 I mean, is zero -zero orthogonal to one -zero? the picture is, you know, zero is just the origin right here and then one -zero is this vector.

01:26 And the answer is yes, they are orthogonal because if you take their drop product, you get zero.

01:31 So the set of vectors where the set where every, single vector is the zero vector is still an orthogonal set.

01:41 So you have to be careful, okay? just because you have an orthogonal set doesn't mean that you can declare it is linearly independent, okay? and then a linearly dependent set cannot be a basis.

01:55 So, of course, so this statement is false, okay? this statement is false and we need to come up with a counter example.

02:04 So some advice for thinking of counter examples, it's usually best to choose the simplest and the most extreme counter example.

02:15 We need to come up with a statement where, so we need to come up with a case where this statement right here fails.

02:23 Okay, this is what it means to have a counter example.

02:28 Okay, well let's take r3.

02:30 Okay.

02:31 And let's wb, so let x1, x2, x3b, so x1b, so x1b, the standard.

02:37 Basis.

02:39 So let x1, x2, x3 be the standard basis.

02:45 Then w is the entire r3, is the entire vector space of r3.

02:50 And there's nothing wrong with doing that.

02:53 Okay.

02:55 So, and then also what will we choose for v1, v2, v3? so let's take v1, v2, v3 to equal the zero vector.

03:07 Okay.

03:12 So this is the simplest counter example i can think of and then the most extreme case where the orthogonality condition fails to guarantee linearly in the linear independence is when you know each of the vectors are the zero vector okay so so x1 x2 x3 are certainly linearly independent okay and v1 v2 v3 well they're definitely orthogonal in w right they're orthogonal r3 all right because i mean this condition right here certainly satisfied right v1 .v2 is zero v1 .v3 is zero and v2 .v3 is equal to zero so that's fine and then we should check that the conclusion actually fails and it certainly does right so in this case even though x1 x2 x3 are linearly independent and even though v1 v2 v3 are orthogonal in w v1 v3 cannot be a basis for w this fails right here simply because you know this is a linearly dependent set okay now we're on to question two all right we're asked if x is not in the subspace w then x minus the projection of x onto w is not zero so this is true okay it's pretty obvious it should be pretty obvious to you that is true right but we can't just you know we can't just wave our hands and justify it, we're going to actually have to come up with a proof.

04:50 Now, this comes from obviously the orthogonal decomposition theorem, right? so you can take any vector x and split it, decompose it into projection onto w plus some vector z, where z is in your orthogonal complement.

05:07 So we're going to use something to do with this.

05:11 All right.

05:13 So let me show you my proof.

05:16 Well, it's a little bit awkward to work with, you know, this x not in a subspace and then this is not zero, right? so let me show you a little mathematical magic trick, if you like.

05:30 So saying a implies b is completely equivalent to saying that not b implies not a.

05:43 Okay.

05:46 If i live in london, then i live in england is equivalent to saying, if i do.

05:51 Don't live in england, then i certainly cannot live in london.

05:55 Right? and so this right here, this fact, we use quite a lot in maths.

06:02 All right.

06:02 I'm going to use this here.

06:04 So instead of showing that x not in a subspace w implies that this is not zero, i'm going to say, okay, well suppose that this guy right here is zero.

06:17 Then i'm going to prove to you that x must be in the subspace.

06:22 Space w.

06:25 This is a little bit confusing...

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