Let R^3 have the Euclidean inner product. Find an orthonormal basis for the subspace spanned by

Let R^3 have the Euclidean inner product. Find an...

Key Concepts

Orthonormal Basis

An orthonormal basis for a subspace is a set of vectors that are both orthogonal (each pair of distinct vectors has a zero inner product) and normalized (each vector has unit length). This type of basis simplifies many computations because it makes projections and coordinate transformations straightforward and minimizes numerical errors.

Gram-Schmidt Process

The Gram-Schmidt process is an algorithm used to transform a set of linearly independent vectors into an orthonormal set that spans the same subspace. It works by iteratively subtracting the projections of each vector onto the previously computed orthogonal vectors followed by normalization, which effectively removes redundancy and dependencies.

Euclidean Inner Product

The Euclidean inner product is a function that takes two vectors and returns a scalar by summing the products of their corresponding components. It establishes a way to define lengths (norms) and angles between vectors, making it essential for calculations related to orthogonality and the geometric interpretation of vector spaces.

Subspace and Span

A subspace of a vector space is a set that is itself a vector space and is closed under vector addition and scalar multiplication. The span of a set of vectors is the smallest subspace containing those vectors, formed by all possible linear combinations, which provides the context for determining the direction and extent of the space covered by the vectors.

Transcript

00:01 So for this exercise we start with a subspace that is spanned by these three vectors.

00:08 V1, v2, and b3.

00:12 Okay, and we need to find an orthonormal basis for this space.

00:18 So the first step is to, well, the point is that we know a procedure that pick a set of vectors, let's say v1 up to vn and then it returns after applying the grannschmidt procedure it returns a set of orthonormal sets an orthonormal set it returns a norm so you start with a set of a set of vectors and you obtain an ortho normal set but to apply this procedure we need to know that these vectors are linearly so before starting to calculate and or applying the grunschmidt procedure for this set of b1, b2 and b3, what we need to do is determine if these vectors are linear independent.

01:14 And to do that, we need to check if one of them is origin as a linear combination of the other one.

01:20 And you can observe that actually the vector b3 is equal to b1 plus b2.

01:29 So you can observe that this just by inspection, but if you want a procedure to check this, like some process or algorithm to do that, what i recommend to do is putting these vectors in a matrix form.

01:51 That means putting the vectors as rows of a matrix.

01:55 That means 0 -1 -2 minus -1 -1.

02:00 One okay and here then you try to reduce this matrix to the action form by applying the gauss procedure and here you will observe that when you want to eliminate this put a zero here or here you will obtain at the end you will only two pi buts one and here something let's say a b here c zero zero zero that means that one of your rows will become full of zeros and that means that you is enough to pick you only two of the vectors that you're considering in this in this set okay if you obtain three pivots after applying the gauss elimination if you obtain three pivots then your three vectors are linearly dependent but in this case what happened is you obtained something of this form or you can check by inspection that the vector b3 is a linear combination of the other two okay so first you determine that if the set is linear independent and we observe that it is not so it's enough to eliminate one of the vectors that you have here in this case i'm going to eliminate b3 and we're going to use b1 and b2 so for this vector for this subspace is enough to write that is the span of the vector v1 and b2 and this set span the subspace w and they are linearly independent so now we can apply the garnesmith procedure to obtain a basis that is ortho -normal okay so let's remember that for the grunschmidt procedure we are going to obtain this set v1 and v2 and after applying the grunschmidt procedure we obtain a set of vectors alpha 1 and alpha 2 both unitary vectors so to do that we need to fix one of the vectors here let's say alpha 1 here i'm used the hat for unitary vectors so this vector here is not unitary so we fix one of the vectors in our set either v1 or b2 to follow the same order i'm going to pick v1 so alpha 1 it's going to be v1 that means the vector 0 1 2 so this is the first step so in our set in our orthogonal set, we have already the vector alpha 1...

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