FoldUnfold Table of Contents The Adjoint of a Linear Map Example 1 The Adjoint of a Linear Map Let $V$ be a vector space over the field $\mathbb{F}$. Recall from the Linear Functionals page that a linear functional is a linear map $\varphi : V \to \mathbb{F}$ from the vector space $V$ into the underlying field, in this case, $\mathbb{F}$. Also recall that if $V$ is a … [Read more...]

## Linear Functionals Examples 1

FoldUnfold Table of Contents Linear Functionals Examples 1 Example 1 Linear Functionals Examples 1 Recall from the Linear Functionals page that if $V$ is a vector space over the field $\mathbb{F}$ then a linear functional $\varphi : V \to \mathbb{F}$ is a linear map from $V$ to the field $\mathbb{F}$. One of the important theorems that we saw is that if $V$ is a … [Read more...]

## Linear Functionals

FoldUnfold Table of Contents Linear Functionals Linear Functionals We will now look at a new type of function that takes vectors $v \in V$ and maps to scalars in $\mathbb{F}$. Definition: A Linear Functional is a linear map $\varphi : V \to \mathbb{F}$ from the vector space $V$ to the field of scalars $\mathbb{F}$. For example, consider the vector space $\mathbb{R}^4$ … [Read more...]

## Orthonormal Vectors Review

FoldUnfold Table of Contents Orthonormal Vectors Review Orthonormal Vectors Review We will now review some of the recent content regarding orthonormal vectors. Recall from the Orthonormal Bases of Vector Spaces page that if $V$ is a finite-dimensional inner product space then an Orthonormal Basis of $V$ is a basis $\{ e_1, e_2, ..., e_n \}$ such that $ = 0$ for all $i, j … [Read more...]

## Norm Minimization Examples 1

FoldUnfold Table of Contents Norm Minimization Examples 1 Example 1 Norm Minimization Examples 1 Recall from the Norm Minimization page that if $V$ is an inner product space and $U$ is a finite-dimensional vector space of $V$ where $V = U \oplus U^{\perp}$, and if we let $v \in V$ then for every vector $u \in U$ we have that: (1) \begin{align} \quad \| v - P_U(v) \| ≤ \| … [Read more...]

## Norm Minimization

FoldUnfold Table of Contents Norm Minimization Norm Minimization Recall from the Orthogonal Projection Operators page that if $V$ is an inner product space and $U$ is a subspace of $V$ such that $V = U \oplus U^{\perp}$ then any $v \in V$ can be written as the sum $v = u + w$ where $u \in U$ and $w \in U^{\perp}$ and the orthogonal projection operator of $V$ onto $U$ is … [Read more...]

## Orthogonal Projection Operators Examples 1

FoldUnfold Table of Contents Orthogonal Projection Operators Examples 1 Example 1 Orthogonal Projection Operators Examples 1 Recall from the Orthogonal Projection Operators page that if $U$ is a subspace of $V$, then $V = U \oplus U^{\perp}$ so that for every vector $v \in V$ we have that $v = u + w$ where $u \in U$ and $w \in U^{\perp}$. We defined the orthogonal … [Read more...]

## Orthogonal Projection Operators

FoldUnfold Table of Contents Orthogonal Projection Operators Orthogonal Projection Operators Recall from the Orthogonal Complements page that if $U$ is a subset of an inner product space $V$, then the orthogonal complement of $U$ denoted $U^{\perp}$ is the set of vectors $v \in V$ such that $v$ is orthogonal to every vector $u \in U$, that is $U^{\perp} = \{ v \in V : = … [Read more...]

## Orthogonal Complements Examples 1

FoldUnfold Table of Contents Orthogonal Complements Examples 1 Example 1 Orthogonal Complements Examples 1 Recall from the Orthogonal Complements page that if $V$ is an inner product space and $U$ is a subset of $V$ then the orthogonal complement of $U$, $U^{\perp}$ is defined to be the subspace of $V$ such that: (1) \begin{align} \quad U^{\perp} = \{ v \in V : = 0 \: … [Read more...]

## Orthogonal Complements

FoldUnfold Table of Contents Orthogonal Complements Orthogonal Complements Definition: Let $V$ be an inner product space., and let $U$ be a subset of vectors from $V$. The Orthogonal Complement of $U$ is the set of vectors $v \in V$ such that $v$ is orthogonal every vector $u \in U$, that is $U^{\perp} = \{ v \in V : \: = 0, \forall u \in U \}$. Take important note … [Read more...]