Example 2 let v 1 2 2 0 find a unit vector u in the same...

Name: example 2 let v 1 2 2 0 find a unit vector u in the same direction as v example 3 let w be the subspace of r2 spanned by x 1 1 find a unit vector z that is a basis for w 83853
Uploaded: 2022-11-10T18:43:13-08:00
Duration: 5 min 38 s
Channel: Joseph David
Description: example 2 let v 1 2 2 0 find a unit vector u in the same direction as v example 3 let w be the subspace of r2 spanned by x 1 1 find a unit vector z that is a basis for w 83853

Example 2: Let v = (1, -2, 2, 0). Find a unit vector u in the same direction as v. Example 3: Let W be the subspace of R² spanned by x = (1, 1). Find a unit vector z that is a basis for W.

Transcript

0:00 Hi.

00:01 In this question, we have a subspace v of r4, which is defined by, i'll just write by the plain w -y, w plus y plus, or sorry, w plus x plus y plus c equals 0.

00:16 So i won't write the whole set, but it's defined by this equation, and it is in fact a linear subspace.

00:25 And then we have vectors, which i'll call a this first vector, and then b, i'll call this first vector, and then b, i'll call the second one.

00:44 So first we want to show that a and b can be part of a basis for v.

00:51 So the first thing that we have to show is that these vectors are in v, right? so we have to show that a hat are in v.

01:07 But this is clear.

01:08 I'll just put a green checkmark because, right, it's clear if you plug this in, you have negative 1 plus 1 to 0.

01:13 So it's clear that a and b are in v.

01:18 The second thing we need to show is just that in order for them to be part of a basis, we need them to be linearly independent.

01:36 Nearily independent.

01:40 And so that can be sometimes a difficult process, or we just have to show that one of them is not a multiple of the other.

01:48 And here, well, we can see that the dot product between them is zero, which implies that they're perpendicular.

01:59 Which implies linear independence.

02:04 So that's an easy way of showing that.

02:07 And then finally we want to expand to a, we want to expand this to a basis for v.

02:16 So we need another vector that's in v, and that's linearly independent from the first two.

02:26 Hi, in this, and so the way i did this actually was just by trying different things, because a and b are sort of nice vectors.

02:37 So we need a vector c such that c is in v, and c is linearly independent of a and b...

Question

Example 2: Let v = (1, -2, 2, 0). Find a unit vector u in the same direction as v. Example 3: Let W be the subspace of R² spanned by x = (1, 1). Find a unit vector z that is a basis for W.

Please give Ace some feedback