We generalize the notion of perpendicularity to [math]\mathbb{R}^n[/math] as follows:[br][br][u]Definition[/u]: Vectors [math]u[/math] and [math]v[/math] in [math]\mathbb{R}^n[/math] are said to be [b]orthogonal[/b] (to each other) if [math]u\cdot v=0[/math], usually denoted by [math]u\perp v[/math].[br][br]Observe that the zero vector is orthogonal to every vector in [math]\mathbb{R}^n[/math].[br][br]It can be shown that for any vectors [math]u[/math] and [math]v[/math] in [math]\mathbb{R}^n[/math], [math]\|u+v\|^2=\|u\|^2+\|v\|^2+2u\cdot v[/math]. If [math]u\perp v[/math], then [math]\|u+v\|^2=\|u\|^2+\|v\|^2[/math], which can be regarded as the [b]Pythagorean theorem[/b] for [math]\mathbb{R}^n[/math].[br][br]By using the Pythagorean theorem repeatedly, we have the following result:[br][br]Suppose [math]w_1, w_2, \ldots, w_p[/math] are [math]p[/math] vectors in [math]\mathbb{R}^n[/math] such that they are orthogonal to each other i.e. [math]w_i\perp w_j[/math] whenever [math]i\ne j[/math]. Then[br][br][math]\|w_1+w_2+\cdots+w_p\|^2=\|w_1\|^2+\|w_2\|^2+\cdots+\|w_p\|^2[/math][br][br][br]In the applet below, you will see why [math]\|u+v\|^2=\|u\|^2+\|v\|^2[/math] is exactly the Pythagorean theorem in [math]\mathbb{R}^2[/math]. Also, the inner product and the angle [math]\alpha[/math] between the two vectors are related by the following formula:[br][br][math]u\cdot v=\|u\|\|v\|\cos(\alpha)[/math][br][br]

Suppose [math]W[/math] is a subspace of [math]\mathbb{R}^n[/math]. A vector [math]z[/math] in [math]\mathbb{R}^n[/math] is said to be [b]orthogonal to[/b] [math]W[/math] if [math]z\perp v[/math] for every [math]v[/math] in [math]W[/math]. [br][br][u]Definition[/u]: The subset [math]W^\perp=\left\{z\in \mathbb{R}^n \ | \ z\perp v \ \text{for all} \ v\in W\right\}[/math] is called the [b]orthogonal complement[/b] of [math]W[/math][br][br]

In the applet below, suppose [math]W=\text{Span}\{w\}[/math]. If [math]w\ne0[/math], [math]W^\perp[/math] is a plane through the origin such that all vectors containing in the plane are orthogonal to [math]w[/math].

In general, give an m x n matrix [math]A[/math], we let [math]r_1, r_2, \ldots,r_m[/math] be [math]m[/math] row vectors in [math]\mathbb{R}^n[/math]. Then consider the homogeneous equation [math]Ax=0[/math]. Suppose [math]v[/math] is in [math]\text{Nul} A[/math] i.e. a solution to this homogeneous equation. Then it is equivalent to saying that [math]r_i\cdot v=0[/math] for [math]i=1,2,\ldots,m[/math] i.e. [math]v[/math] is orthogonal to [math]\text{Row} A[/math]. Hence, we have[br][br][math](\text{Row} \ A)^\perp=\text{Nul} \ A[/math][br][br]As the above result is true for all matrices, we apply it to [math]A^T[/math] and get [br][br][math](\text{Row} \ A^T)^\perp=\text{Nul} \ A^T[/math][br][br]It is obvious that [math]\text{Row} \ A^T=\text{Col} \ A[/math]. Therefore, we have[br][br][math](\text{Col} \ A)^\perp=\text{Nul} \ A^T[/math]