A的核空间和A的行向量构成的空间互为补空间吗 求一道矩阵的零空间和列空间的维数的问题
\u9f50\u6b21\u65b9\u7a0b\u7ec4\u89e3\u7684\u95ee\u9898\uff1f\u7406\u89e3(1) \u7528\u884c\u53d8\u6362\u5f97\u7cfb\u6570\u77e9\u9635A\u7684\u4e3a\u79e9 r ( A) = r \uff0cA \u81f3\u5c11\u6709\u4e00\u4e2a r \u9636\u5b50\u5f0f\u4e0d\u4e3a0 \u2192 \u2014\u2014\u2192 \u9f50\u6b21\u7ebf\u6027\u65b9\u7a0b\u7ec4 Ax = 0 \u76f8\u5e94\u7684 r \u4e2a\u65b9\u7a0b\u76f8\u4e92\u72ec\u7acb\u3002 \u2014\u2192 \u4e00\u4e2a\u65b9\u7a0b\u53ef\u4ee5\u89e3\u51fa\u4e00\u4e2a\u672a\u77e5\u91cf\u3002r\u4e2a\u76f8\u4e92\u72ec\u7acb\u7684\u65b9\u7a0b\u53ea\u80fd\u89e3\u51fa r \u4e2a\u672a\u77e5\u91cf\u3002\u65b9\u7a0b\u7ec4\u7684\u901a\u89e3\u4e2d\u5fc5\u5b9a\u542b\u6709 n-r \u4e2a\u72ec\u7acb\u672a\u77e5\u91cf\u3002 \u7406\u89e3(2) \u8fd9 n-r \u4e2a\u81ea\u7531\u672a\u77e5\u91cf\u7684\u6bcf\u4e00\u7ec4\u503c\uff0c\u552f\u4e00\u5730\u4ea7\u751f\u51fa\u4e00\u4e2a\u89e3\u5411\u91cf\u3002\u65b9\u7a0b\u7ec4 Ax = 0 \u7684\u4e24\u4e2a\u4e0d\u76f8\u7b49\u7684\u89e3\u5411\u91cf\uff0c\u5fc5\u5b9a\u5bf9\u5e94\u7740\u4e0d\u540c\u7684\u4e24\u7ec4\u81ea\u7531\u672a\u77e5\u91cf\u503c\u3002n-r \u4e2a\u81ea\u7531\u672a\u77e5\u91cf\u6309\u7167\u5176\u539f\u5e8f\u6392\u6210\u6709\u5e8f\u7ec4\uff0c\u5c31\u662f n-r \u7ef4\u5411\u91cf\u3002\u8fd9\u5c31\u662f\u8bf4\uff0c\u9f50\u6b21\u7ebf\u6027\u65b9\u7a0b\u7ec4 Ax = 0 \u7684\u89e3\u5411\u91cf\u96c6\u5408\u4e0e\u5168\u4f53 n-r \u7ef4\u5411\u91cf\u6210\u529f\u4e00\u4e00\u5bf9\u5e94\u3002 \u7406\u89e3(3) \u6309\u7167\u539f\u5e8f\uff0c\u9010\u6b21\u53d6 n-r \u4e2a\u81ea\u7531\u672a\u77e5\u91cf\u4e3a n-r \u7ef4\u7684\u5355\u4f4d\u5411\u91cf\u7ec4\uff0c\u4ee3\u56de\u90a3 r \u4e2a\u65b9\u7a0b\uff0c\u7b97\u5f97r\u4e2a\u672a\u77e5\u91cf\u7684\u503c\u3002\u518d\u8fd4\u56de\u5c06n-r \u7ef4\u7684\u5355\u4f4d\u5411\u91cf\u7ec4\u201c\u5ef6\u957f\u201d\u4e3a\uff08n \u7ef4\uff09\u89e3\u5411\u91cf\u7ec4\u3002 \u201c\u7ebf\u6027\u65e0\u5173\uff0c\u5ef6\u957f\u65e0\u5173\u201d\u3002 \u201c\u5ef6\u957f\u201d\u6240\u5f97\u89e3\u5411\u91cf\u7ec4\u5c31\u662f\u57fa\u7840\u89e3\u7cfb\u3002 \u56e0\u6b64\u8bf4\uff0c\u201c\u5982\u679c\u7cfb\u6570\u77e9\u9635A\u7684\u79e9\u4e3ar (A)\uff0c\u5219\u9f50\u6b21\u7ebf\u6027\u65b9\u7a0b\u7ec4Ax = 0\u5177\u6709 n-r \u7ef4\u7684\u7684\u89e3\u7a7a\u95f4\u3002\u201d\uff08\u6f5c\u53f0\u8bcd\uff1an \u7ef4\u5411\u91cf\u7a7a\u95f4\u7684\u4e00\u4e2a n - r \u7ef4\u5b50\u7a7a\u95f4\u3002\uff09 \u901a\u5e38\uff0c\u6211\u4eec\u53ef\u4ee5\u7c97\u7cd9\u5730\u8bf4\uff0c\u9f50\u6b21\u7ebf\u6027\u65b9\u7a0b\u7ec4 Ax = 0\u89e3\u96c6\u7684\u79e9 = n-r (A\uff09 \u8fd9\u5c31\u662f\u201c\u9f50\u6b21\u7ebf\u6027\u65b9\u7a0b\u7ec4\u89e3\u96c6\u6784\u9020\u7406\u8bba\u201d\u3002 \u5b83\u8d2f\u7a7f\u4e8e\u300a\u7ebf\u6027\u4ee3\u6570\u300b\u6559\u6750\u59cb\u7ec8\u3002\u5c24\u5176\u5728\u7279\u5f81\u7406\u8bba\u57fa\u7840\u90e8\u5206\u8981\u53cd\u590d\u8fd0\u7528\u3002\u662f\u8003\u7814\u6570\u5b66\u5e74\u5e74\u5fc5\u8003\u7684\u8003\u70b9\u3002\u5f88\u9057\u61be\u7684\u662f\uff0c\u591a\u6570\u56fd\u5185\u672c\u79d1\u6559\u6750\u5e76\u6ca1\u6709\u7a81\u51fa\u8fd9\u4e00\u70b9\u3002 \u65b9\u6cd5\u4e8c\uff1a\u7528\u66f4\u9ad8\u5c42\u6b21\u7684\u65b9\u6cd5\u53ef\u4ee5\u8bc1\u660e\uff0c\u57fa\u7840\u89e3\u7cfb\u662f\u9f50\u6b21\u7ebf\u6027\u65b9\u7a0b\u7ec4 Ax = 0\u89e3\u96c6\u7684\u6700\u5927\u65e0\u5173\u7ec4\u3002 \u5982\u679c\u5185\u79ef \u03b1\u03b2\u02ca= 0\uff0c\u79f0\u5411\u91cf \u03b1 \u4e0e \u03b2 \u6b63\u4ea4\u3002 \u5229\u7528\u6b63\u4ea4\u6982\u5ff5\uff0c\u53ef\u4ee5\u7ed9\u9f50\u6b21\u7ebf\u6027\u65b9\u7a0b\u7ec4 Ax = 0 \u7684\u89e3\u5411\u91cf\u4e00\u4e2a\u65b0\u7684\u542b\u610f\uff1a \u5411\u91cf \u03b1 \u662f\u9f50\u6b21\u7ebf\u6027\u65b9\u7a0b\u7ec4 Ax = 0 \u7684\u89e3\u5411\u91cf\u7684\u5145\u5206\u5fc5\u8981\u6761\u4ef6\u662f\uff0c\u03b1 \u4e0e A \u7684\u6bcf\u4e2a\u884c\u5411\u91cf\u90fd\u6b63\u4ea4\u3002 \u8fd9\u6837\u4e00\u6765\uff0cA \u7684 r \u4e2a\u7ebf\u6027\u65e0\u5173\u7684\u884c\u5411\u91cf\u751f\u6210\u7684\u5b50\u7a7a\u95f4\u4e0e\u89e3\u5411\u91cf\u7a7a\u95f4\u4e92\u4e3a\u6b63\u4ea4\u8865\u7a7a\u95f4\u3002\u6545\u89e3\u5411\u91cf\u7a7a\u95f4\u7684\u79e9\u53ea\u80fd\u662fn-r \u5373 \u57fa\u7840\u89e3\u7cfb\u662f\u9f50\u6b21\u7ebf\u6027\u65b9\u7a0b\u7ec4 Ax = 0\u89e3\u96c6\u7684\u6700\u5927\u65e0\u5173\u7ec4\u3002(\u5185\u79ef\u4e0e\u6b63\u4ea4\u6027\u8bf7\u53c2\u770b\u6211\u7684\u8003\u7814\u6570\u5b66\u8bb2\u5ea7\uff0840\uff09\u5411\u91cf\u5185\u79ef\u5b66\u5728\u524d)[]
\u5b8c\u5168\u641e\u9519\u4e86\uff0c\u5217\u7a7a\u95f4\u7ef4\u6570\u5176\u5b9e\u5c31\u662f\u77e9\u9635\u7684\u79e9\uff0c\u56e0\u4e3a\u6709\u96f6\u5411\u91cf\uff0c\u7528\u8116\u5b50\u60f3\u4e5f\u4e0d\u53ef\u80fd\u662f3\u554a\uff01\u6613\u77e5\u5269\u4e0b\u4e24\u4e2a\u5411\u91cf\u4e0d\u5171\u7ebf\uff0c\u6240\u4ee5\u8bf4\u79e9\u662f2\uff0c\u96f6\u7a7a\u95f4\u7ef4\u6570\u662f1\u3002\u3002\u3002
是的，核空间就是A的零空间（AX=0的解空间），与行空间（行向量构成的子空间）互为补空间绛旓細鏄殑锛鏍哥┖闂灏辨槸A鐨勯浂绌洪棿锛圓X=0鐨勮В绌洪棿锛夛紝涓琛岀┖闂达紙琛屽悜閲忔瀯鎴鐨勫瓙绌洪棿锛変簰涓鸿ˉ绌洪棿
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