Prove that each of the following formulas for ⟨𝐯, 𝐰⟩defines an inner product on ℝ^3. Verify all

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Prove that each of the following formulas for ⟨𝐯, 𝐰⟩defines...

Prove that each of the following formulas for $\langle\mathbf{v}, \mathbf{w}\rangle$ defines an inner product on $\mathbb{R}^3$. Verify all the inner product axioms in careful detail: (a) $v_1 w_1+2 v_2 w_2+3 v_3 w_3$, (b) $4 v_1 w_1+2 v_1 w_2+2 v_2 w_1+4 v_2 w_2+v_3 w_3$, (c) $2 v_1 w_1-2 v_1 w_2-2 v_2 w_1+3 v_2 w_2-v_2 w_3-v_3 w_2+2 v_3 w_3$.

Prove that each of the following formulas for $\langle\mathbf{v}, \mathbf{w}\rangle$ defines an inner product on $\mathbb{R}^3$. Verify all the inner product axioms in careful detail:
(a) $v_1 w_1+2 v_2 w_2+3 v_3 w_3$,
(b) $4 v_1 w_1+2 v_1 w_2+2 v_2 w_1+4 v_2 w_2+v_3 w_3$,
(c) $2 v_1 w_1-2 v_1 w_2-2 v_2 w_1+3 v_2 w_2-v_2 w_3-v_3 w_2+2 v_3 w_3$.

Key Concepts

Positive Definiteness and Quadratic Forms

Positive definiteness requires that the inner product of any nonzero vector with itself is positive, and zero only for the zero vector. When an inner product is represented in coordinate form, it gives rise to a quadratic form, and the condition of positive definiteness implies that the associated quadratic form is strictly positive for non-zero inputs. This condition ensures that the inner product defines a norm that genuinely measures length.

Symmetry

Symmetry is the property that the inner product yields the same result regardless of the order of its arguments. In mathematical terms, for any two vectors v and w in the space, the inner product satisfies ?v, w? = ?w, v?. This is critical because it ensures that the geometry encoded by the inner product, such as the angle between vectors, is well-defined.

Matrix Representation of Bilinear Forms

In finite-dimensional spaces, an inner product can be represented as a matrix transformation. The coefficients in the given formulas can be arranged into a symmetric matrix, with the inner product of two vectors being expressed as v^T A w. Verifying that the matrix A is symmetric and positive definite provides a systematic way of confirming that the given formula defines a valid inner product. This method connects the abstract axioms of inner products to concrete computational techniques using matrices.

Inner Product Axioms

An inner product on a vector space must satisfy certain axioms: linearity in the first argument (which implies bilinearity when combined with symmetry), symmetry (the inner product is invariant under switching the two vectors), and positive definiteness (the inner product of a vector with itself is strictly positive unless the vector is the zero vector, in which case it is zero). These properties ensure that the inner product behaves in a way that allows for geometric interpretation, including notions of length and angle.

Bilinearity (Linearity in Each Argument)

Bilinearity means the inner product is linear in each of its arguments when the other is held fixed. That is, it obeys additivity and homogeneity in both slots. This property is fundamental as it allows the decomposition of computations involving the inner product into sums of more basic scalar operations, making the structure of the vector space more tractable and linking the inner product to matrix operations.

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