Gilbert Strang-Linear Algebra-Orthogonality_null space and the measurement space are orthogona

作者：你好赵伟 | 2024-07-16 11:43:28

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null space and the measurement space are orthogonal.

Orthognality

Orthogonality of the four spaces

Def.1 Orthognality:
Two subspaces $V$ and $W$ of a vector space are $Orthogonal$ if every vector $\pmb{v}$ is perpendicular to every vector $\pmb{w}$ in $\pmb{W}$ .
Orthogonal subspaces

v v^{T} w w = 0

$\pmb{v}^T \pmb{w} = 0$ for all

v v

$\pmb{v}$ in

V V

$\pmb{V}$ and

w w

$\pmb{w}$ in

W W

$\pmb{W}$ .

The surface of the floor and the wall is perpendicular, but these two spaces are not orthogonal, since we can still find two vectors in each space that are not perpendicular.
Null space and Row Space are orthogonal
If $\pmb{x}\in N(A)$ , then we have $A\pmb{x} = 0$ .

[\begin{matrix} r o w_{1} \\ r o w_{2} \\ . . . \\ r o w_{m} \end{matrix}] [\begin{matrix} x \end{matrix}] = [\begin{matrix} r o w_{1} ∙ x \\ r o w_{2} ∙ x \\ . . . \\ r o w_{m} ∙ x \end{matrix}] = [\begin{matrix} 0 \\ 0 \\ . . . \\ 0 \end{matrix}]

$\begin{bmatrix} row_1\\row_2\\...\\row_m\end{bmatrix}\begin{bmatrix}x\end{bmatrix}=\begin{bmatrix}row_1 \bullet x\\row_2 \bullet x \\ ...\\row_m \bullet x\end{bmatrix}=\begin{bmatrix}0\\0\\...\\0\end{bmatrix}$
Inner product of

x x

$\pmb{x}$ and the matrix

A

$A$ equals

0

$0$ . Similiarly we can prove that Row space and left null space are orthogonal.

Orthognal Complemen

Row space and Null space split $\pmb{R^n}$ into two orthogonal subspaces. For example for matrix $A=\begin{bmatrix}1&2&5\\2&4&10\end{bmatrix}$ ,

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