Linear Algebra with Applications, 10th Edition

5.2 Orthogonal Subspaces

Let A be an $m\times n$ matrix and let $\mathbf{x}\in N\left(A\right)$, the null space of A. Since $A\mathbf{x}=\mathbf{0}$, we have

$$a_{i1}{x}_1+a_{i2}{x}_2+\mathrm{\dots }+a_{in}{x}_n=0$$ (1)

for $i=1,\mathrm{\dots },m$. Equation (1) says that x is orthogonal to the ith column vector of $A^T$ for $i=1,\mathrm{\dots },m$. Since x is orthogonal to each column vector of $A^T$, it is orthogonal to any linear combination of the column vectors of $A^T$. So if y is any vector in the column space of $A^T$, then ${\mathbf{x}}^T\mathbf{y}=0$. Thus, each vector in N(A) is orthogonal to every vector in the column space of $A^T$. When two subspaces of ${ℝ}^n$ have this property, we say that they are orthogonal.

Definition

Two subspaces X and Y of ${ℝ}^n$ are said to be orthogonal if ${\mathbf{x}}^T\mathbf{y}=0$ for every $\mathbf{x}\in X$ and every $\mathbf{y}\in Y$. If X and Y are orthogonal, we write $X\perp Y$.

Example 1

Let X be the ...