Prove that if 𝐯1, 𝐯2 form a basis of an inner product space V and 𝐯1=𝐯2, then 𝐯1+𝐯2 and 𝐯1-𝐯2 fo

Prove that if 𝐯1, 𝐯2 form a basis of an inner product space...

Prove that if $\mathbf{v}_1, \mathbf{v}_2$ form a basis of an inner product space $V$ and $\left\|\mathbf{v}_1\right\|=\left\|\mathbf{v}_2\right\|$, then $\mathbf{v}_1+\mathbf{v}_2$ and $\mathbf{v}_1-\mathbf{v}_2$ form an orthogonal basis of $V$.

Key Concepts

Properties of the Inner Product

Key properties of the inner product include symmetry (or conjugate symmetry in complex spaces), linearity in each argument, and positive-definiteness. These properties are used to establish important relationships between vectors, particularly to demonstrate that the inner product of certain combinations of vectors is zero, which is the definition of orthogonality.

Norm

The norm of a vector is a measure of its length and is defined using the inner product. It is used to quantify the size or magnitude of vectors in the space. In the context of the problem, the fact that the two basis vectors have the same norm plays a crucial role in proving the orthogonality of their sum and difference.

Basis

A basis of a vector space is a set of vectors that are linearly independent and span the entire space. Working with different bases can simplify the analysis of vector properties and transformations, as well as provide a framework for constructing and understanding new bases, such as an orthogonal basis.

Orthogonal Basis

An orthogonal basis is a basis consisting of vectors that are mutually perpendicular according to the inner product defined on the space. Having such a basis is valuable because it simplifies computations in the space, especially those involving lengths, projections, and decomposition of vectors into components along the directions of the basis vectors.

Inner Product Space

An inner product space is a vector space equipped with an additional structure called an inner product. This inner product allows for the definition of concepts such as length (or norm) and angle between vectors, making it possible to discuss orthogonality, projections, and related geometric properties within the space.

Transcript

0:00 Hello there.

00:01 So for this exercise we're working with an inner product space v that have a basis b sub b defined by the vectors b1, b2 up to vr.

00:14 And we need to prove that the only vector that is orthogonal to this space b is the zero vector.

00:20 So for that let's first consider so let's consider a subspace of b defined as the set of all the vectors w in this in the in the inner product space b such that the inner product of w with v i is equal to zero for all the v i's in the basis of b so we need to show that this subset has uh is a is a only only contains the zero vector okay so basically is to say that w is a singlet so let's show this so first we need to show that this set is non empty and we know by definition of the inner product that the zero vector the inner product of the zero vector with any vector b is equals to zero for every v on the on the space so this is the general definition of of the inner property of the inner product and this implies that then the zero vector is also equals to zero the inner product of the zero vector with any vector in the basis it's going to be equals to zero for every vi in the basis so we can say that the zero vector is an element on w and this implies that w is not which is great but then we need to show that w has an as an only element the dog the zero vector so for that we need to consider a generic vector in in the subspace w so let's consider so any vector in w due to w is a subspace of b is a equivalent to say that w can be written as a linear combination from i up to r of the basis elements okay because if w is an element of the subspace w then w is also an element of the vector space and if w is an element of the vector space b then this can be expressed as a linear combination of the basis elements so then we can consider the inner product of w with itself.

03:34 And we are going to express this w as a linear combination of the basis elements of b...

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