线性代数;设4维列向量a1,a2,a3线性无关且与4维列向量b1,b2均正交,证明b1,b2线性相关 线性代数;设4维列向量a1,a2,a3线性无关且与4维列向量...
\u7ebf\u6027\u4ee3\u6570;\u8bbe4\u7ef4\u5217\u5411\u91cfa1,a2,a3\u7ebf\u6027\u65e0\u5173\u4e14\u4e0e4\u7ef4\u5217\u5411\u91cfb1,b2\u5747\u6b63\u4ea4,\u8bc1\u660eb1,b2\u7ebf\u6027\u76f8\u5173\u4f60\u597d\uff0c\u5f88\u9ad8\u5174\u4e3a\u60a8\u89e3\u7b54
\u8bc1\u660e\uff1a\u5411\u91cf\u7ec4a1,a2,a3,b1,b2\u4e00\u5b9a\u7ebf\u6027\u76f8\u5173\uff0c\u6240\u4ee5\u5b58\u5728\u4e0d\u5168\u4e3a\u96f6\u7684\u5b9e\u6570x1,x2,x3,y1,y2\u4f7f\u5f97x1a1+x2a2+x3a3+y1b1+y2b2=0\uff0c\u5373x1a1+x2a2+x3a3=-y1b1-y2b2\u3002
\u5219b1,b2\u4e0d\u80fd\u5168\u4e3a\u96f6\uff0c\u5426\u5219x1a1+x2a2+x3a3=0\uff0c\u56e0\u4e3aa1,a2,a3\u7ebf\u6027\u65e0\u5173\uff0c\u6240\u4ee5x1,x2,x3\u5168\u4e3a\u96f6\uff0c\u6240\u4ee5x1,x2,x3,y1,y2\u5168\u4e3a\u96f6\uff0c\u77db\u76fe\u3002
\u6240\u4ee5y1,y2\u4e0d\u80fd\u5168\u4e3a\u96f6\u3002
a1,a2,a3\u4e0eb1,b2\u90fd\u6b63\u4ea4\uff0c\u6240\u4ee5x1a1+x2a2+x3a3\u4e0eb1,b2\u90fd\u6b63\u4ea4\uff0c\u6240\u4ee5x1a1+x2a2+x3a3\u4e0ey1b1+y2b2\u6b63\u4ea4\uff0c\u6240\u4ee5(x1a1+x2a2+x3a3,y1b1+y2b2)=(-(y1b1+y2b2),y1b1+y2b2)=-y1b1+y2b2,y1b1+y2b2)=0\uff0c\u6240\u4ee5y1b1+y2b2=0\u3002
\u56e0\u4e3ay1,y2\u4e0d\u5168\u4e3a\u96f6\uff0c\u6240\u4ee5b1,b2\u7ebf\u6027\u76f8\u5173\u3002
\u5e0c\u671b\u80fd\u591f\u5e2e\u52a9\u5230\u4f60\uff0c\u671f\u5f85\u60a8\u7684\u91c7\u7eb3\uff0c\u8c22\u8c22
\u4ee5a1,a2,a3\u7684\u8f6c\u7f6e\u4e3a\u884c\u5411\u91cf\u6784\u9020\u65b9\u7a0b\u7ec4Ax=0\uff0c\u5219\u5411\u91cfb1,b2\u90fd\u662f\u65b9\u7a0b\u7ec4Ax=0\u7684\u89e3\u3002Ax=0\u67093\u4e2a\u65b9\u5411\uff0c4\u4e2a\u672a\u77e5\u91cf\uff0c\u56e0\u4e3aa1,a2,a3\u7ebf\u6027\u65e0\u5173\uff0c\u6240\u4ee5A\u7684\u79e9r(A)=3\uff0c\u6240\u4ee5Ax=0\u7684\u57fa\u7840\u89e3\u7cfb\u91cc\u9762\u67094-3=1\u4e2a\u5411\u91cf\u3002
b1,b2\u90fd\u662fAx=0\u7684\u89e3\uff0c\u53ef\u7531Ax=0\u7684\u57fa\u7840\u89e3\u7cfb\u7ebf\u6027\u8868\u793a\uff0c\u6240\u4ee5r(b1,b2)\u22641\uff0c\u6240\u4ee5b1,b2\u7ebf\u6027\u76f8\u5173\u3002
由于a1,a2,a3线性无关，则a1,a2,a3张成一个三维空间Q，由于a1,a2,a3与b1,b2均正交，即Q与b1,b2张成的空间正交，若b1,b2线性无关，则a1,a2,a3,b1,b2会张成一个五维空间，与它们是四维向量矛盾。
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绛旓細|-2A|=(-2)^4*|A|= 32
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绛旓細=(-2)^4*2 =32
绛旓細杩欎笉鏄樉鐒剁殑鍚 浠=[伪,尾]锛孉=XX^T锛宺ank(A)<=rank(X)<=2
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