Which of the following define inner products on ℂ^2 ? (a) ⟨𝐯, 𝐰⟩=v1 w̅1+2 i v2 w̅2, (b) ⟨𝐯, 𝐰⟩=v

step 1 First, let's check property 1. It's easy to see that this property is satisfied. And next, the s

Which of the following define inner products on ℂ^2 ? (a)...

Which of the following define inner products on $\mathbb{C}^2$ ? (a) $\langle\mathbf{v}, \mathbf{w}\rangle=v_1 \bar{w}_1+2 \mathrm{i} v_2 \bar{w}_2$, (b) $\langle\mathbf{v}, \mathbf{w}\rangle=v_1 w_1+2 v_2 w_2$, (c) $\langle\mathbf{v}, \mathbf{w}\rangle=v_1 \bar{w}_2+v_2 \bar{w}_1$, (d) $\langle\mathbf{v}, \mathbf{w}\rangle=$ $2 v_1 \bar{w}_1+v_1 \bar{w}_2+v_2 \bar{w}_1+2 v_2 \bar{w}_2$, (e) $\langle\mathbf{v}, \mathbf{w}\rangle=2 v_1 \bar{w}_1+(1+\mathrm{i}) v_1 \bar{w}_2+(1-\mathrm{i}) v_2 \bar{w}_1+3 v_2 \bar{w}_2$.

Which of the following define inner products on $\mathbb{C}^2$ ? (a) $\langle\mathbf{v}, \mathbf{w}\rangle=v_1 \bar{w}_1+2 \mathrm{i} v_2 \bar{w}_2$,
(b) $\langle\mathbf{v}, \mathbf{w}\rangle=v_1 w_1+2 v_2 w_2$,
(c) $\langle\mathbf{v}, \mathbf{w}\rangle=v_1 \bar{w}_2+v_2 \bar{w}_1$,
(d) $\langle\mathbf{v}, \mathbf{w}\rangle=$
$2 v_1 \bar{w}_1+v_1 \bar{w}_2+v_2 \bar{w}_1+2 v_2 \bar{w}_2$,
(e) $\langle\mathbf{v}, \mathbf{w}\rangle=2 v_1 \bar{w}_1+(1+\mathrm{i}) v_1 \bar{w}_2+(1-\mathrm{i}) v_2 \bar{w}_1+3 v_2 \bar{w}_2$.

Key Concepts

Inner Product Definition

An inner product on a vector space is a function that assigns each pair of vectors a scalar and must satisfy three properties: linearity (or sesquilinearity in the complex case), conjugate symmetry, and positive definiteness. This definition ensures that the inner product can be used to define geometric concepts such as length and angles.

Sesquilinearity

In a complex vector space, the inner product is required to be linear in one argument and conjugate linear in the other. This means that while one argument behaves in the standard linear way, the other argument includes the complex conjugate when scaling, ensuring the operation results in a proper complex scalar that adheres to the structure of the space.

Conjugate Symmetry

Conjugate symmetry is a fundamental property of complex inner products which states that swapping the two vectors in the inner product yields the complex conjugate of the original value. Formally, for any two vectors u and v, the relation ?u, v? = overline{?v, u?} must hold. This property is crucial to maintaining consistency with the geometric and algebraic structure of complex spaces.

Positive Definiteness

Positive definiteness requires that the inner product of any nonzero vector with itself is strictly positive, while it is zero if and only if the vector itself is the zero vector. This property is essential for defining a norm on the vector space, allowing the inner product to measure lengths and ensure a meaningful notion of distance.

Hermitian Matrix Representation

In finite-dimensional complex vector spaces, an inner product can be represented by a Hermitian matrix. The matrix must satisfy that it is equal to its own conjugate transpose and must be positive definite. This representation is useful in verifying whether a given bilinear or sesquilinear form qualifies as an inner product, as it translates the abstract properties into concrete matrix conditions.

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