投影1-正交投影

一个向量在向量上的投影#card=math&code=P_a%28v%29&id=dzMvu)，是%5Chat%7Ba%7D#card=math&code=%28v%2C%20%5Chat%7Ba%7D%29%5Chat%7Ba%7D&id=jI8Rj), 这里, 即与同向的单位向量。由于 %22%20aria-hidden%3D%22true%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-28%22%20x%3D%220%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-76%22%20x%3D%22389%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-2212%22%20x%3D%221097%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%3Cg%20transform%3D%22translate(2097%2C0)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-50%22%20x%3D%220%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20transform%3D%22scale(0.707)%22%20xlink%3Ahref%3D%22%23E1-MJMATHI-61%22%20x%3D%22908%22%20y%3D%22-213%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-28%22%20x%3D%223214%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-76%22%20x%3D%223604%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-29%22%20x%3D%224089%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-2C%22%20x%3D%224479%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%3Cg%20transform%3D%22translate(4924%2C0)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-61%22%20x%3D%2213%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJSZ1-2C6%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-29%22%20x%3D%225481%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-3D%22%20x%3D%226148%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-28%22%20x%3D%227204%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-76%22%20x%3D%227594%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-2C%22%20x%3D%228079%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%3Cg%20transform%3D%22translate(8524%2C0)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-61%22%20x%3D%2213%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJSZ1-2C6%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-29%22%20x%3D%229081%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-2212%22%20x%3D%229692%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-28%22%20x%3D%2210693%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-76%22%20x%3D%2211083%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-2C%22%20x%3D%2211568%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%3Cg%20transform%3D%22translate(12013%2C0)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-61%22%20x%3D%2213%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJSZ1-2C6%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-29%22%20x%3D%2212570%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-28%22%20x%3D%2212959%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%3Cg%20transform%3D%22translate(13349%2C0)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-61%22%20x%3D%2213%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJSZ1-2C6%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-2C%22%20x%3D%2213905%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%3Cg%20transform%3D%22translate(14351%2C0)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-61%22%20x%3D%2213%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJSZ1-2C6%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-29%22%20x%3D%2214907%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-3D%22%20x%3D%2215574%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-30%22%20x%3D%2216631%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3C%2Fsvg%3E#card=math&code=%28v-P_a%28v%29%2C%20%5Chat%7Ba%7D%29%20%3D%20%28v%2C%20%5Chat%7Ba%7D%29-%28v%2C%20%5Chat%7Ba%7D%29%28%5Chat%7Ba%7D%2C%20%5Chat%7Ba%7D%29%3D0&id=tJEz4),
故#card=math&code=a-P_a%28v%29&id=zVaqH)与向量正交，于是

)%5Cperp%20P_a(v).%0A#card=math&code=%28a-P_a%28v%29%29%5Cperp%20P_a%28v%29.%0A&id=wzj9y)

进一步，如果是中的-平面，则任意一个向量%5E%5Ctop#card=math&code=v%3D%28v_1%2C%20v_2%2C%20v_3%29%5E%5Ctop&id=Dgkk3)在上的投影是%5E%5Ctop#card=math&code=%28v_1%2C%20v_2%2C%200%29%5E%5Ctop&id=fNTAe).

推而广之，如果是有限维实欧几里得空间，是非零子空间，记是的正交补子空间。如果#card=math&code=%28e1%2C%20%5Cldots%2C%20e_n%29&id=SrwRg)是的一个标准正交基，我们可以将它扩充为的一个标准正交基#card=math&code=%28e_1%2C%20%5Cldots%2C%20e_n%2C%20e%7Bn%2B1%7D%2C%20%5Cldots%2C%20e%7Bk%7D%29&id=CXirg), 其中. 则任意一个向量%5E%5Ctop%20%3D%5Csum%7Bi%3D1%7D%5Ek%20vi%20e_i#card=math&code=v%3D%28v_1%2C%20%5Cldots%2C%20v_k%29%5E%5Ctop%20%3D%5Csum%7Bi%3D1%7D%5Ek%20vi%20e_i&id=bzKuY)，在与 上的投影分别是.

问题在于，如果的一个标准正交基还不知道，怎么计算投影？答案是，用Gram-Schmidt正交化，先”制造”出上的一个标准正交基，然后再用3中的方法计算投影。

从投影到子空间 的映射是满射。向量在上的投影，记作#card=math&code=P_W%28v%29&id=NfEuK), 可以由矩阵乘法给出：%3DPv#card=math&code=P_W%28v%29%3DPv&id=unbRG), 其中被成为从到的投影矩阵。可以证明投影是线性映射，于是，只要知道的一个基被投影到哪些矩阵，就能求出。由直和分解, 以及投影的定义，可知中任何向量，投影到中，仍然得到这个向量本身。同理，对中的向量，也是如此。所以在3中提到的那个标准正交基之下，矩阵是阶单位矩阵。