Prove that the following define Hermitian inner products on ℂ^2 : (a) ⟨𝐯, 𝐰⟩=v1 w̅1+2 v2 w̅2, (b

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Prove that the following define Hermitian inner products on...

Prove that the following define Hermitian inner products on $\mathbb{C}^2$ :
(a) $\langle\mathbf{v}, \mathbf{w}\rangle=v_1 \bar{w}_1+2 v_2 \bar{w}_2$,
(b) $\langle\mathbf{v}, \mathbf{w}\rangle=v_1 \bar{w}_1+\mathrm{i} v_1 \bar{w}_2-\mathrm{i} v_2 \bar{w}_1+2 v_2 \bar{w}_2$.

Key Concepts

Hermitian Inner Product

A Hermitian inner product is a function on a complex vector space that assigns a complex number to each pair of vectors, satisfying the properties of conjugate symmetry, linearity (or sesquilinearity), and positive definiteness. This structure generalizes the notion of a dot product to accommodate complex vector spaces.

Sesquilinearity

Sesquilinearity means that the inner product is linear (or additive and homogeneous) in one argument and conjugate-linear (or anti-linear) in the other. This property ensures that scaling and addition behave correctly across the complex field, a crucial aspect when defining and working with inner products in complex spaces.

Conjugate Symmetry

Conjugate symmetry is the requirement that the inner product of two vectors reverses when the arguments are swapped, specifically that the inner product of v with w equals the complex conjugate of the inner product of w with v. This property is fundamental in ensuring that the inner product returns a real number when a vector is paired with itself.

Positive Definiteness

Positive definiteness implies that the inner product of any nonzero vector with itself is strictly positive, thereby ensuring a meaningful notion of length or norm in the space. This property is essential for applications that rely on measuring distances and angles in complex vector spaces.

Quadratic Forms

When an inner product is applied to a vector with itself, the resulting expression is a quadratic form. Analyzing these quadratic forms helps verify the positive definiteness of the inner product, ensuring that the norm derived from it behaves as expected.

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