# 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

for $i=1,\mathrm{\dots},m$. Equation (1) says that **x** is orthogonal to the *i*th 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 ${\mathbb{R}}^{n}$ have this property, we say that they are orthogonal.

# Example 1

Let *X* be the ...

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