# Adjoint of a linear map

Aug-Nov 2020

## Recap

• Vector space $V$ over a scalar field $F$
• $F$: real field $\mathbb{R}$ or complex field $\mathbb{C}$ in this course
• $m\times n$ matrix A represents a linear map $T:F^n\to F^m$
• dim null $T$ + dim range $T$ = dim $V$
• Solution to $Ax=b$ (if it exists): $u+$ null$(A)$
• Four fundamental subspaces of a matrix
• Column space, row space, null space, left null space
• Eigenvalue $\lambda$ and Eigenvector $v$: $Tv=\lambda v$
• Some linear maps are diagonalizable
• Inner products, norms, orthogonality and orthonormal basis
• Upper triangular matrix for a linear map over an orthonormal basis
• Orthogonal projection gives closest vector in the subspace
• Least squares solution to a linear equation is orthogonal projection

## Recall: Linear functionals and inner product

$V$: finite-dimensional inner product space over $F=\mathbb{R}$ or $\mathbb{C}$

$\phi:V\to F$ is a linear functional

Riesz representation theorem: There exists unique $u\in V$ such that $\phi(v)=\langle v,u\rangle$

How to find $u$ for a given $\phi$?

Orthonormal basis for $V$: $e_1,\ldots,e_n$

$u=\overline{\phi(e_1)}e_1+\cdots+\overline{\phi(e_n)}e_n$

## Linear functionals from a linear map

$V,W$: finite-dimensional inner product space over $F=\mathbb{R}$ or $\mathbb{C}$

$T:V\to W$ is a linear map

Fix some $w\in W$

Let $\phi_{T,w}(v)=\langle Tv,w\rangle$

$\phi_w:V\to F$ is a linear functional

Riesz: There exists unique $u_{T,w}\in V$ s.t. $\phi_{T,w}(v)=\langle Tv,w\rangle=\langle v,u_{T,w}\rangle$

## Example

$x=(x_1,x_2,x_3,x_4)\in\mathbb{R}^4$

$Tx=(x_1+2x_2+3x_3+4x_4,3x_1+4x_2+5x_3+6x_4)\in\mathbb{R}^2$

1. $w=(1,2)$

$\langle Tx,w\rangle=7x_1+10x_2+13x_3+16x_4$

$u_{T,w}=(7,10,13,16)$

1. $w=(w_1,w_2)$

$\langle Tx,w\rangle=(w_1+3w_2)x_1+(2w_1+4w_2)x_2+(3w_1+5w_2)x_3+(4w_1+6w_2)x_4$

$u_{T,w}=(w_1+3w_2,2w_1+4w_2,3w_1+5w_2,4w_1+6w_2)\in\mathbb{R}^4$

## Adjoint of a linear map

$V,W$: finite-dimensional inner product space over $F=\mathbb{R}$ or $\mathbb{C}$

$T:V\to W$ is a linear map

For $w\in W$, let $u_{T,w}\in V$ be s.t.

$\langle Tv,w\rangle=\langle v,u_{T,w}\rangle$

The adjoint of $T$, denoted $T^*$, is the function from $W$ to $V$ mapping $w$ to $u_{T,w}$

$T^*$: adjoint of $T$ satisfies

$\langle Tv,w\rangle=\langle v,T^*w\rangle$

## Example

$x=(x_1,x_2,x_3,x_4)\in\mathbb{R}^4$

$Tx=(x_1+2x_2+3x_3+4x_4,3x_1+4x_2+5x_3+6x_4)\in\mathbb{R}^2$

$w=(w_1,w_2)\in\mathbb{R}^2$

$\langle Tx,w\rangle=(w_1+3w_2)x_1+(2w_1+4w_2)x_2+(3w_1+5w_2)x_3+(4w_1+6w_2)x_4$

$T^*w=(w_1+3w_2,2w_1+4w_2,3w_1+5w_2,4w_1+6w_2)\in\mathbb{R}^4$

## Adjoint is a linear map

$V,W$: finite-dimensional inner product space over $F=\mathbb{R}$ or $\mathbb{C}$

$T:V\to W$ is a linear map

$T^*$: adjoint of $T$ is a linear map from $W$ to $V$

Proof

\begin{align} \langle v,T^*(w_1+w_2)\rangle &= \langle Tv,w_1+w_2\rangle\\ &=\langle Tv,w_1\rangle+\langle Tv,w_2\rangle\\ &=\langle v,T^*w_1\rangle+\langle v,T^*w_2\rangle\\ &=\langle v,T^*w_1+T^*w_2\rangle \end{align}

\begin{align} \langle v,T^*(aw)\rangle &= \langle Tv,aw\rangle\\ &=\bar{a}\langle Tv,w\rangle\\ &=\bar{a}\langle v,T^*w\rangle\\ &=\langle v,aT^*w\rangle \end{align}