5.2 Orthogonal Subspaces

Let A be an m×n matrix and let xN(A), the null space of A. Since Ax=0, we have

ai1x1+ai2x2++ainxn=0 (1)

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

Example 1

Let X be the ...

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