Prove that the only element 𝐰 in an inner product space V that is orthogonal to every vector, so

Prove that the only element 𝐰 in an inner product space V...

Key Concepts

Inner Product Space

An inner product space is a vector space equipped with an inner product, a function that assigns a scalar to each pair of vectors and satisfies properties such as linearity, symmetry (or conjugate symmetry in complex spaces), and positive-definiteness. This setting allows for the definition of geometric concepts such as length and angle, forming the foundation for many analytical techniques.

Orthogonality

Orthogonality is the concept where two vectors are considered perpendicular if their inner product is zero. This idea generalizes the familiar perpendicularity in Euclidean spaces to abstract vector spaces, and it underlies methods in projections, decompositions, and constructions of orthogonal bases.

Nondegeneracy

Nondegeneracy of the inner product asserts that if the inner product of a vector with every vector in the space is zero, then that vector must be the zero vector. This property is crucial in ensuring that the inner product meaningfully reflects the structure of the space and is fundamental in proofs establishing that only the zero vector can be orthogonal to all vectors.

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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