Dual Bases: Given a basis 𝐯1, …, 𝐯n of V, the dual basis ℓ1, …, of V^* consists of the linear fu

step 1 Okay, so we're looking at question 12 .4, and we're asked to prove the theorem of theorem 12 .1.

Dual Bases: Given a basis 𝐯1, …, 𝐯n of V, the dual basis ℓ1,...

Dual Bases: Given a basis $\mathbf{v}_1, \ldots, \mathbf{v}_n$ of $V$, the dual basis $\ell_1, \ldots, \ell_n$ of $V^*$ consists of the linear functions uniquely defined by the requirements $\ell_i\left(\mathbf{v}_j\right)= \begin{cases}1 & i=j, \\ 0, & i \neq j .\end{cases}$ (a) Show that $\ell_i[\mathbf{v}]=x_i$ gives the $i^{\text {th }}$ coordinate of a vector $\mathbf{v}=x_1 \mathbf{v}_1+\cdots+x_n \mathbf{v}_n$ with respect to the given basis. (b) Prove that the dual basis is indeed a basis for the dual vector space. (c) Prove that if $V=\mathbb{R}^n$ and $A=\left(\mathbf{v}_1 \mathbf{v}_2 \ldots \mathbf{v}_n\right)$ is the $n \times n$ matrix whose columns are the basis vectors, then the rows of the inverse matrix $A^{-1}$ can be identified as the corresponding dual basis of $\left(\mathbb{R}^n\right)^*$.

Dual Bases: Given a basis $\mathbf{v}_1, \ldots, \mathbf{v}_n$ of $V$, the dual basis $\ell_1, \ldots, \ell_n$ of $V^*$ consists of the linear functions uniquely defined by the requirements $\ell_i\left(\mathbf{v}_j\right)= \begin{cases}1 & i=j, \\ 0, & i \neq j .\end{cases}$
(a) Show that $\ell_i[\mathbf{v}]=x_i$ gives the $i^{\text {th }}$ coordinate of a vector $\mathbf{v}=x_1 \mathbf{v}_1+\cdots+x_n \mathbf{v}_n$ with respect to the given basis. (b) Prove that the dual basis is indeed a basis for the dual vector space. (c) Prove that if $V=\mathbb{R}^n$ and $A=\left(\mathbf{v}_1 \mathbf{v}_2 \ldots \mathbf{v}_n\right)$ is the $n \times n$ matrix whose columns are the basis vectors, then the rows of the inverse matrix $A^{-1}$ can be identified as the corresponding dual basis of $\left(\mathbb{R}^n\right)^*$.

Key Concepts

Matrix Representation and Inversion

In finite-dimensional vector spaces, the representation of basis vectors in a matrix form enables a clear connection between the vector space and its dual. Specifically, when the basis vectors are organized as columns of an invertible matrix, the rows of the inverse matrix correspond to the dual basis functionals. This relationship not only provides an algebraic method to compute the dual basis but also illustrates the deep connection between linear transformations and their inverses in the context of duality.

Coordinate Functions

Coordinate functions are linear maps that extract the coefficients (coordinates) of a vector when expressed in terms of a given basis. They are naturally represented by the dual basis elements and are fundamental in understanding how vectors are represented in various coordinate systems, allowing the abstract vector to be handled concretely via its coordinates.

Basis

A basis of a vector space is a set of vectors that is both linearly independent and spans the entire space. This fundamental concept is key in linear algebra, as it allows every element of the space to be uniquely represented as a linear combination of the basis vectors, serving as the building blocks for the vector space.

Dual Vector Space

The dual vector space of a vector space V, denoted V*, is the collection of all linear functions (or linear functionals) from V to its field of scalars. This concept is crucial because it provides a way to study vectors through the action of linear maps, and it plays a significant role in many areas of mathematics including functional analysis and differential geometry.

Dual Basis

Given a basis of a vector space, the dual basis consists of linear functionals that are defined to pick out the coordinates corresponding to each basis vector. In other words, each dual basis vector acts on the original basis vectors in a way that produces the Kronecker delta, which is essential for transitioning between the space and its dual and for performing coordinate calculations.

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