Prove that the formula ⟨𝐯, 𝐰⟩=v1 w1-v1 w2-v2 w1+b v2 w2 defines an inner product on ℝ^2 if and o

step 1 First, we solve part B. The inner product is defined as this expression, and we want to know whe

Prove that the formula ⟨𝐯, 𝐰⟩=v1 w1-v1 w2-v2 w1+b v2 w2...

Key Concepts

Inner Product

An inner product is a function that takes two vectors from a vector space and returns a scalar while satisfying specific properties: linearity in the first argument (and, in the real case, symmetry which implies linearity in the second argument as well), symmetry (or conjugate symmetry in complex spaces), and positive definiteness. These conditions ensure that the inner product can be used to define lengths (norms) and angles, providing the geometric structure necessary for many applications in mathematics and related fields.

Bilinear Form

A bilinear form is a function that is linear in each of its arguments independently. In the context of inner products, the function must be bilinear, which means that it respects both additivity and homogeneity in each variable separately. Recognizing that the given formula is a bilinear form helps in applying linear algebraic techniques to analyze its properties such as symmetry and positive definiteness.

Symmetry

Symmetry in the context of inner products means that the product of two vectors does not depend on the order of the vectors; that is, the inner product of v with w equals the inner product of w with v. This property is essential for ensuring that the geometric interpretation of the inner product is consistent, and it plays a critical role in proving or disproving whether a given bilinear form qualifies as an inner product.

Positive Definiteness

Positive definiteness is a key property of an inner product, requiring that the inner product of any non-zero vector with itself is strictly positive. In quadratic form language, this means that the associated quadratic form has only positive values (except at the zero vector). For the given bilinear form to serve as an inner product, conditions on its parameters must enforce this property, leading to constraints such as b > 1.

Matrix Representation and Determinant Conditions

Every bilinear form can be represented in matrix form, where the inner product corresponds to a quadratic form involving a symmetric matrix. The positive definiteness of this matrix can be determined by checking whether all its eigenvalues are positive, or equivalently, by ensuring that its leading principal minors (determinants) are positive. In the case of a 2×2 matrix, these conditions translate to having a positive leading diagonal entry and a positive determinant, which in turn provides the necessary condition (b > 1) for the bilinear form to be an inner product.

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