[線代] orthogonal projection
在黃子嘉的習題本做到這題:
[ 1 2 2 ] [ 1 ]
Let A = [ 0 0 1 ] and v = [ 1 ], and let U be the column space of A.
[ 1 0 2 ] [ 0 ]
Find a vector u0 in U such that ||v - u0|| = min||v - u|| for all u in U.
這題基本上是找orthogonal projection, 假設u0 = Ax0,
x0要滿足: (A^t)Ax0 = (A^t)v, where A^t is the transpose of A.
稍作整理:
[ 2 4 2 ] [ 1 ]
(A^t)A = [ 4 8 4 ], (A^t)v = [ 2 ]
[ 2 4 5 ] [ 3 ]
因此在求xo的時候,
是否可以看成在解一個nonhomogeneous system of linear equations?
要將xo表達成"s + K_H"的形式嗎? (請參閱Friedberg的書)
s是指滿足nonhomogenenous system的解, K_H是相對應的homogeneous system的解集合
因為我看黃子嘉的解答是令x0等於s, 再左乘以A, 即得到u0.
這邊是我有疑惑的地方, 為什麼不用表達成"s + K_H"的形式?
謝謝
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